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These are the classes that have been submitted for teaching on Saturday and Sunday. Please note that teachers' plans evolve and there is no guarantee that any specific class listed here will be taught.

### Turtle

Level: High Intermediate

Creator: Beth Johnson

Teacher: Beth Johnson

Description:
Turtle based on a tessellation pattern.

Special Requirements: I will be bringing pre-cut hexagon shaped paper.

### Fountain Pen

Level: Intermediate

Creator: Hiroshi Kumasaka

Description:
This model is folded from 8-10 inch square foil.

### Abstract Stingray

Level: Intermediate

Creator: Amy Goncharsky

Teacher: Amy Goncharsky

Description:
A simple box-pleated ray that curves naturally. Mostly reverse-folds, but the body requires some three-dimensional shaping.

### Mouse

Level: High Intermediate

Creator: Masashi Tanaka

Teacher: James Peake

Description:
Super adorable mouse! Really comes to life with the shaping.

### Gay's Practical Purse

Level: Low Intermediate

Creator: Gay Merrill Gross

Teacher: Elizabeth Detmold

Description:
There's perhaps no model that I've folded more over the years than this one. It has a multitude of uses: Wallet, sewing kit, holder for whatever doodads--even a prom purse. Let your imagination guide you! It includes a reverse fold--hence some might not consider it a beginner's model--but I do. It's a good model to learn a few things on, and all are welcome! The purse looks especially nice folded in duo paper, and a wide variety of papers can be used. Duo scrapbooking paper will be provided.

### Lion

Level: High Intermediate

Creator: Rob Snyder

Teacher: James Peake

Description:
A cute lion with a stylized mane, characteristic of Rob Snyder's geometric animals. Each student will receive a complimentary sheet of 14in Deluxe Washi for the class (pictured). Bonus model taught with any extra time!

Special Requirements: Involves shaping!

### Origami Fiery Phoenix

Level: Complex

Creator: Malcolm Smith

Teacher: Malcolm Smith

Description:
This a complex model with many pleats and repeated steps.The tail is especially detailed and may require extra shaping after or during the class.

Special Requirements: A good thumb for tessellating.

### Folded Origami Greetings

Level: Low Intermediate

Creator: Various

Teacher: Edith Kort

Description:
We will fold a variety of greeting cards. These include pop-out map folds inserted into a rectangular card, origami decorations attached to a traditional card, a folded card with a place for a message inside, and a fold that is integrated into the card design.

Special Requirements: Designs will be folded from a variety of papers (some non-square) provided by the instructor. Some require the use of adhesive.

### Infinite Twist

Level: Intermediate

Creator: Goran Konjevod

Teacher: Goran Konjevod

Description:
Pleat tessellation that naturally twists into a double helix.

### Loggerhead Sea Turtle

Level: Complex

Creator: Justin Fossum and Alexander Miller

Teacher: Justin Fossum

Description:
This is a sea turtle with back plates.

Special Requirements: Students should be able to fold tiny pleats, know how to open and close sink. Students should also be able to do squash, swivel and spread folds. Students will need to crimp to shape the head.

### Flapping Dragon

Level: Complex

Creator: Justin Fossum

Teacher: Justin Fossum

Description:
This is an action model of a dragon based on the bird base.

Special Requirements: Student needs to be able to fold rabbit ears, open sink, crimp, pleat and have general knowledge of the bases.

### Origami Stars

Level: Low Intermediate

Creator: Sok Song

Teacher: Sok Song

Description:
Various easy modular stars!

### Wen's Dragon

Level: Intermediate

Creator: Sok Song

Teacher: Sok Song

Description:
Whimsical and sweet, this dragon was created for a dear friend who always sang Puff the Magic Dragon whenever we went to go Karaoke!

Special Requirements: the model is easy, but be warned there is an open sink. its not too hard though, it uses a bird base.

### Dollar Rose

Level: High Intermediate

Creator: Sok Song

Teacher: Sok Song

Description:
Dollar bill rose with stem and leaf.

Special Requirements: This model is a little tricky because of the small size of a dollar and it requires a lot of small intricate folds. Be prepared to fold small or bring a larger dollar-sized paper.

### Soffrey Puppy

Level: Low Intermediate

Creator: Sok Song

Teacher: Sok Song

Description:
Cute and easy puppy with an adorable oversized head! And a fun color change! This model name is also a combination mashup of names Sok+Jeffrey=Soffrey!

### Scolli Pig

Level: Intermediate

Creator: Sok Song

Teacher: Sok Song

Description:
Simple and easy to fold Pig! This is one of my favorites, I designed it for two very good friends of mine Scott and Olli, that is why its called the Scolli Pig.

### 2 Piece Triangle Star + Bonus

Level: Simple

Creator: Lewis Simon and Charlene Morrow

Teacher: Charlene MORROW

Description:
This star is made from 2 equilateral triangles, which will be provided. Time permitting, I will teach how to make an equilateral triangle from a square. It's an easy model to fold once you have the triangles and very clever in the way the two units interconnect. This is a very satisfying model for the beginner - or anyone! As a bonus, I will teach a very easy sailboat (pictured) that I designed with the leftover paper from making the equilateral triangles. The Star's creator is Lewis Simon; the Sailboat's, Charlene Morrow.

### Hungry Shark

Level: High Intermediate

Creator: Jon Tucker

Teacher: Jon Tucker

Description:
An action model based on a variation of the traditional hungry crow/hungry fox model.

Level: Intermediate

Teacher: Jon Tucker

Description:
This is an intermediate level mask based on the mask for The Lion King movie. It is designed by Joe Adia, @joeygami on Instagram.

Special Requirements: Paper recommendation: 10" kami, Tant, or similar

### Simple Animals Workshop

Level: Simple

Creator: Andrew Hudson

Teacher: Andrew Hudson

Description:
We will fold simple animals until we run out of time! For the past couple years I have been focusing on creating and teaching models with folding sequences that are elegant and memorable. These designs have been playtested on 6-year-olds but are sophisticated enough that more experienced folders will still enjoy them.

### Smile!

Level: High Intermediate

Creator: Masaya Horiguchi

Teacher: Kayo Kurata

Description:
This model is folded with 1 piece of 6-inch regular origami paper or duo color. It is all straight lines and lots of intricate folds.

### Star Behind Star (1-sheet)

Level: Intermediate

Creator: Jacques Justin

Teacher: Deb Pun

Description:
You'll learn how to fold this versatile, fabulous 1-sheet model – no cutting, no gluing! A 6-inch square yields about a 3-inch star suitable for greeting cards, origami pins and holiday decoration. I will teach you my folding sequence to fold an 8x8 grid aligning cut edges and folded edges of the paper as a guide. After precreasing an 8x8 grid, we will need to carefully form into a 3D shape and then twist-collapse the star. I will share my variation that displays the major color on the front and back sides of the star. The original diagram shows the major color on the front and a minor color on the back. Diagrams will be distributed.

Special Requirements: Accurate pre-creasing and careful shaping is helpful for the model. Great folded from kami, Korean pattern paper, patterned duo, Grimmhobby Corona Harmony print, Aitoh Bokashi Chiyogami flower pattern print, Tuttle Rainbow Pattern print (https://cdn11.bigcommerce.com/s-q39b4/images/stencil/2000x2000/products/5495/215626/9780804846363.in01__41592.1488892908.jpg).

### Origami Star Box

Level: Intermediate

Creator: Yuki Honma

Teacher: Pamela Miike

Description:
Folded from A-4 paper: one sheet yields two boxes. It can be a puffy box or a flat tato-like envelope. Precise folding is needed to ensure an easy collapse into this box shape. It is designed by Yuki Honma of Sweet Paper.

### Modular Star

Level: Intermediate

Creator: Enrica Dray

Teacher: Janet Yelle

Description:
Modular Star by Enrica Dray is created from 8 squares. The modules are simple and the assembly is intermediate. It’s a fun, versatile star that has dimension but can also lie flat. Lots of beautiful results can be attained with different papers and variations. It is pretty on a card, as a decoration and can even be used a a pin.

### Small Flower Octagon Box

Level: Low Intermediate

Creator: Tomoko Fuse

Teacher: Anjana V

Description:
In this class, we'll be making an octagonal box that's functional and very useful for storing things--like a pack of playing cards. This design comes from the book Fabulous Origami Boxes by Tomoko Fuse. The box is 5" wide and 1" tall. The lid and bottom each are made up of 4 pieces. We will make all 8 pieces and assemble them.

Special Requirements: Some prior experience with modular origami would be helpful. 8 sheets of 6" kami are required for this model

### Maneki Neko (Lucky Cat)

Level: High Intermediate

Creator: Oriol Esteve

Teacher: Oriol Esteve

Description:
This is a self-standing Japanese lucky cat that will bring you good fortune. The 15 cm (6-inch) version can easily hold a coin in its belly.

Special Requirements: Kami red/white or black/red works best.

### Hyperbolic Modular Unit

Level: Intermediate

Creator: William Holt

Teacher: William Holt

Description:
Using an adapted hyperbolic paraboloid unit, we can make an origami approximation of something called a Triply Periodic Minimal Surface -- specifically the Schwarz's D Surface. You don't have to know that in order to fold it, and once you're done it will still look spectacular! Also holds bananas (not provided)

Special Requirements: Ideally you should know how to fold a hyperbolic paraboloid, or a saddle shape, but this is not required. There are 6 units required for the basic model, 18 units required for the more complex model which will be explained but not taught, with a potential for an unlimited number of units. We will start with 4 square sheets and cut the paper to six sheets of the correct size.

### Duck Duck Goose

Level: Intermediate

Creator: Brian Webb

Teacher: Brian Webb

Description:
Is it a duck or a goose, you decide. Action model developed over the past few years that has defined reference points. Best folded from 15cm paper of a single color, like tant, but plain old kami or printer paper works too.

### It’s a Windy Day!

Level: Low Intermediate

Creator: Anne LaVin

Teacher: Anne LaVin

Description:
We’ll use this easy piece - a sailboat which works well with either regular or thicker paper, dampened - to introduce wetfolding and practice handling and constructing in 3D.

### Rotational Star Tessellation

Level: Supercomplex

Creator: Omer Shalev

Teacher: Omer Shalev

Description:
Well worked out tessellation from one square. A hit at OUSA back in June; will be teaching it again at PCOC. Stars within a circular shape as well as central flaps or a central star. Has a nice sequence as well as a neat backside to it. Twist fold and precreasing accuracy encouraged. We will go as far the general gist of the model as time permits. Class can finish the model at their free will as we may not finish. I will also teach it again at hospitality free folding area if interest shown, as well as the other variations.

Special Requirements: Must be comfortable with several precise foldings, 22.5 and twist folds, there being a sequence of landmark folds to be done before collapse.

### Tsuru Noshi

Level: Intermediate

Creator: Unknown

Teacher: Andrew Hans

Description:
A crane with a noshi as part of the wings. Uses duo paper.

Special Requirements: Must be familiar with how to fold a crane using the "star" method inscribed on the paper to collapse it to the bird base. Will be using a Robert Lang technique to find a reference point to start the fold. Uses a spot of glue to hold down the strips of paper for the noshi.

Level: Complex

Creator: Michael Assis

Teacher: Michael Assis

Description:
A 3D tessellation lampshade will be taught starting from a pre-folded triangular grid. The folded triangular grid should be on a square piece of paper with divisions of 24 or 32.

Special Requirements: Each person must bring with them a pre-folded triangular grid folded on a square piece of paper with divisions of 24 or 32. Familiarity with tessellations / corrugations is not necessary but preferable. Students on-site will be given a sheet of 12-inch tant to work with. 10-inch kami can be used but is not preferable.

### Pentagon Vase

Level: Intermediate

Creator: Arnold Tubis

Teacher: Arnold Tubis

Description:
Class participants will learn how to fold/cut a regular pentagon from a square and then fold a five-pointed vase with the inner section framed by a regular pentagram. The design can be easily generalized to similar vases from triangles, squares, and hexagons.

Special Requirements:

### Duck

Level: High Intermediate

Creator: Jared Needle

Teacher: Jared Needle

Description:
It's a duck. Oregon likes ducks right? Small detailed folds and a closed unwrap make this a bit trickier than it looks, but it's worth it.

### Crane Shaped Receptacle

Level: Simple

Creator: Noriko Sumida

Teacher: Arlene Gorchov

Description:
Cute crane that has a little space where you could put a small chocolate or candy. Paper will be provided for the class.

### Foalie the Pony

Level: High Intermediate

Creator: Michael LaFosse

Teacher: Arlene Gorchov

Description:
Delightful action model pony that makes a clip clip sound when folded from really crisp paper. Paper will be provided for the class.

Level: Intermediate

Creator: Joel Stern

Teacher: Joel Stern

Description:
This African Mask is made from ordinary letter paper, is 3-dimensional, and its facial features move when you stretch the model.

Special Requirements: This model is made from 8 1/2 x 11 (letter size) paper which will be provided for the class.

### Paper Puppetry

Level: Low Intermediate

Creator: Various

Teacher: Ramsay Cowlishaw

Description:
Learn to fold a handful (pun intended) of simple paper puppets and then learn basic puppetry techniques showing how to bring them to life. One puppet is from "Trick Origami" by Yoshide Momotani; others are from "Folding Paper Puppets" by Shari Lewis & Lillian Oppenheimer.

Special Requirements: There will be a cut for one of the models but it will be fine if students tear the paper. No need for scissors. Materials will be provided for the class.

### Modular Box

Level: Low Intermediate

Creator: Tomoko Fuse and Traditional

Teacher: Holly Williams

Description:
4-piece modular box lid by Tomoko Fuse and masu bottom. My regular students have loved making this design in multiple sizes. Great for gift giving!

Special Requirements: The box lid is described in this link: http://www.origami-instructions.com/modular-square-box-p1.html. The bottom is the standard masu box. We will use 4 pieces of 2 sided 15 cm paper for the lid and an 8.5" square for the bottom.

### African elephant

Level: Complex

Creator: Robby Kraft

Teacher: Robby Kraft

Description:
New African elephant design with a color change for the tusks.

Special Requirements: paper larger than 6 inches is best. be okay with folding off the table and shaping in 3D

### Fennec Fox

Level: Simple

Creator: Toshikazu Kawasaki

Teacher: Toshikazu Kawasaki

Description:
This is an improved (in 2016) version of Fennec Fox from The Little Prince -- in the book おとぎの国の夢折り紙

### Naruto Whirlpools

Level: Simple

Creator: Toshikazu Kawasaki

Teacher: Toshikazu Kawasaki

Description:
This is a modular origami with movable joints. It is versatile in terms of size and number of sheets of paper. 4 sheets of 6" square is standard.

### Daffodil Net

Level: Intermediate

Creator: Toshikazu Kawasaki

Teacher: Toshikazu Kawasaki

Description:
This is a specially arranged version of Daffodil from the book Origami Dream World: Flowers & Animals Edition. It was designed in 2017.

Special Requirements: Please bring 2 sheets of 6" kami in green and a cutting tool. Yellow kami for the flowers will be provided. Rectangular strips will be cut.

### Armor Ball

Level: High Intermediate

Creator: Toshikazu Kawasaki

Teacher: Toshikazu Kawasaki

Description:
It is not easy to guess how this model is structured from its appearance. It is a very sturdy sphere.

Special Requirements: Prior to the class, please fold 6 pieces of bird base out of 6" or larger kami. Fold 2 each of 3 colors.

### Lizard

Level: Intermediate

Creator: Toshikazu Kawasaki

Teacher: Toshikazu Kawasaki

Description:
This is an improved (in 2016) version of Lizard from the book Origami Dream World: Flowers & Animals Edition. Folded out of one square.

Special Requirements: Prior to the class, please fold 8x8 valley grids on a 6-10 inch square of heavier weight paper such as tant. Use a larger sheet of paper unless you are a precise folder. Tant or similar weight paper 6-inch will be provided for the class

### Desert Rose

Level: Intermediate

Creator: Toshikazu Kawasaki

Teacher: Toshikazu Kawasaki

Description:
This modular model reflects the fun of paper overlaps. There are many arrangements you can make with this model. Most recently designed in August 2019.

Special Requirements: Please bring 6" squares. You need multiples of 6 sheets. Standard is 12 sheets. No glue.

### One-Minute Rose

Level: Complex

Creator: Toshikazu Kawasaki

Teacher: Toshikazu Kawasaki

Description:
This is the most challenging version of Kawasaki Rose since there are no reference points. Once you master it perfectly, you can fold it in one minute.

Special Requirements: 6-inch tant or similar weight is best; paper will be provided for the class

### Wearable Origami Crane Hat

Level: Intermediate

Creator: Tim Rickman

Teacher: Timothy Rickman

Description:
Learn how to fold this unique Origami Crane Hat. Impress your friends! Be the talk at parties! (Large paper will be provided in class, however, if you would like to bring your own paper for a wearable hat, make sure the side length of your square is 1.366 times the measurement of your head circumference.)

Special Requirements: The most difficult steps of this model are re-folding a crease partially within a pocket, inserting a flap into a pocket, and rabbit ear folds. If you can fold a traditional origami crane nicely you should be able to fold this model.

### Fuse Box

Level: Intermediate

Creator: Tomoko Fuse

Teacher: MaeDean Erb

Description:
Many boxes are created with self closing lids but this took an amazing, mathematical insight as she perfected this jewel. This lovely box is challenging fun.

### Bloomsome

Level: High Intermediate

Creator: Meenakshi Mukerji

Teacher: Meenakshi Mukerji

Description:
A never before taught or published modular design. We will not finish in a 45 minute class but you will have enough knowledge to finish at home or in after hours.

Special Requirements: Must have experience with modulars.

### Clover Heart

Level: High Intermediate

Creator: Meenakshi Mukerji inspired by Fujimoto. Ow

Teacher: Meenakshi Mukerji

Description:
In honor and memory of Shuzo Fujimoto and Francis Ow, inspired by their work.

Special Requirements: Must be familiar with the Shuzo Fujimoto's Hydrangea collapse. Red-green Duo paper is recommended though Kami is OK for learning.

### Easy Flashers

Level: Intermediate

Creator: Jeremy Shafer

Teacher: Malachi Brown

Description:
A couple of Jeremy Shafer's easier flasher designs. Very fun action models to play with once they are dry. Materials will be provided for the class.

### Tangram 2

Level: Simple

Creator: Francis Ow

Teacher: Annette Hatch

Description:
This model uses scissors to create smaller squares or rectangles which are then folded. The tangram pieces are not just cut from a square.

Special Requirements: This model will use 4 pieces of paper. Scissors and/or paper slicers may be borrowed.

### Carambola Flower

Level: Intermediate

Creator: Carmen Sprung

Teacher: Malachi Brown

Description:
Carmen Sprung's delightful flower folded from a pentagon. Paper will be provided for the class.

### Letterfolds

Level: Intermediate

Creator: Various

Teacher: Maria Velazquez

Description:
Elephant letterfold - Michel Grand; Random Envelope - Swamy; more if time permits

Special Requirements: A4 paper will be provided for the class.

### Mr. Fish

Level: Complex

Creator: João Charrua

Teacher: João Charrua

Description:
The goal is to create an extraordinary imaginary creature. Despite its simple look, it has some unconventional steps.

Special Requirements: Students must be comfortable working in 3D

### Mette Rings

Level: Low Intermediate

Creator: Mette Pederson

Teacher: Jan Polish

Description:
This brilliant, versatile wreath by Mette Pederson makes a great pin, ornament, or even a picture frame, depending on the type or size paper you use. You can also add your own creative flourishes to make variations. Paper will be provided for the class.

### Winged Venus

Level: High Intermediate

Creator: João Charrua

Teacher: João Charrua

Description:
The goal is to create an abstract sculptural model from a traditional base using tension techniques.

Special Requirements: Elephant hide sheets work best (30 cm, 12-inches)

### Picasso Tear and Walking Cube

Level: Intermediate

Creator: João Charrua

Teacher: João Charrua

Description:
Two models will be taught in this class. One is completely 2D with a colour change and the other is full 3D. Two simple and fun models are offered so you can have a good time.

Special Requirements: You have to be fan of Cubism.

### From Silhouette to 3D Mask

Level: Intermediate

Creator: João Charrua

Teacher: João Charrua

Description:
The goal is to teach a model that starts as a very simple base and work it until it gets a 3d shape. The class may have a freestyle shaping part at the end.

Special Requirements: This is a class that requires a certain level of shaping skills and lots of creativity :)

### Scoring the Curved Flower

Level: Intermediate

Creator: Ekaterina Lukasheva

Teacher: Ekaterina Lukasheva

Description:
Instead of traditional folding, we will score the pattern onto the paper by hand and then fold it.

Special Requirements: A scoring tool or old ballpoint/gel pen that does not write anymore. I have some, but please bring yours! A hard surface or small cutting mat (preferred) -- 15cm would be enough. A ruler. Accuracy :)

### Pillows Tessellation

Level: High Intermediate

Creator: Ekaterina Lukasheva

Teacher: Ekaterina Lukasheva

Description:
We are going to fold a tessellation!

Special Requirements: Having folded some tessellations before, understanding that there's some skill involved. Some small clips would be useful.

### Greater Kudu

Level: Complex

Creator: Robert J. Lang

Teacher: Robert Lang

Description:
This is a complex design that requires precise folding and close attention to reference points. There are complex open and closed sink folds and closed wraps; you should be comfortable with all of these. It works well wet-folded from shapeable paper, but we'll use inside-out foil for ease of teaching.

Special Requirements: There are no diagrams, but you're welcome to take photos of step folds along the way.

### Octagonal Star Pot

Level: High Intermediate

Creator: Robert J. Lang

Teacher: Robert Lang

Description:
This is a 3D pot folded from a square, High Intermediate in difficulty. The hardest individual steps are merely squash folds, but there will be tiny tabs going into tiny pockets, and if you don't fold crisply and precisely, the model will blow up in the final step; only take this class if you're a precise folder.

### Spring Flower

Level: Intermediate

Creator: Marcia Joy Miller

Teacher: Katherine Riley

Description:
Intermediate model, participants should know the water bomb base and be comfortable doing sinks. We will work with copy paper. Materials will be provided for the class.

### Cross

Level: Intermediate

Creator: Matthew Green

Teacher: Kathy Stevick

Description:
Lovely cross -- especially nice when using mandala paper. Paper will be provided for the class.

Special Requirements: Precise folding and sharp creases are required.

### Five-Petal Flower #1

Level: Low Intermediate

Creator: Alexandra Abovyan

Teacher: Maureen Miller-Calamo

Description:
This is a modular flower. Five squares are folded and attached with a little bit of glue. There is an exciting action as you open up the attached modules. Then a few finishing touches produces the 3D flower.

### Dice Tower

Level: High Intermediate

Creator: Jason Hathaway

Teacher: Jason Hathaway

Description:
A functional origami dice tower for use in board games and tabletop role-playing games. I will provide thick scrapbooking paper for the class.

### Super Nobu Unit Modular

Level: Intermediate

Creator: Nobuko Okabe

Teacher: Nobuko Okabe

Description:
There will be several modular models all of which can be made with the same simple unit. Different assembly methods will be shown in the class and participants can choose what they would like to make. Because of this versatility of the unit, Ms. Tomoko Fuse named the unit "Super Nobu Unit."

Special Requirements: 3-inch paper will be provided for the class

### Flower

Level: Intermediate

Creator: Ekaterina Lukasheva

Teacher: Ekaterina Lukasheva

Description:
Curved flower folded from a pentagon. Paper will be provided for the class.

### Terrarium Flower

Level: Intermediate

Creator: Yara Yagi

Teacher: Rachel Katz

Description:
This delightful one-piece model works well for origami jewelry, package decoration or a holiday ornament. It is folded from a pentagon which we will create in class.

### Bottlenose Dolphin

Level: High Intermediate

Creator: Henry Tien

Teacher: Henry Tien

Description:
Learn how to fold an intermediate dolphin with eyes.

### Star Festival

Level: High Intermediate

Creator: Nobuko Okabe

Teacher: Nobuko Okabe

Description:
This is a 16 piece modular model. Folding the units is very simple, but the assembly requires experience in modular origami.

### Wearable Bat-Man Helmet

Level: High Intermediate

Creator: naveen rutgers

Teacher: naveen rutgers

Description:
In this class we will fold a 3-dimensional batman mask out of one (preferably large) square piece of paper. the finished product can be worn if folded from appropriately-sized paper.

Special Requirements: Students should know how to do spread-sinks and closed sinks. Stiffer paper that holds its shape is recommended. If you want to wear the finished product, a piece of paper about 80cmx80cm (32-inches) is recommended. Materials will be provided for the class.

### Origami Money Folds

Level: Intermediate

Creator: Arnold Tubis

Teacher: Arnold Tubis

Description:
I will teach several models that will be in a sequel to my book, Tipping with Origami Money Folds/A novel and easy way to show your appreciation of good service.

Special Requirements: Bring to class crisp dollar bills (real or fake) and small clips to hold the pieces of a model together during assembly. I will have a limited number of crisp fake bills and clips to distribute.

### Roaring Lion

Level: Complex

Creator: Joel Garcia Moix

Teacher: Joel Garcia Moix

Description:
A fierce lion that roars. It has the mane and tail tip with change of color. Do you dare to face this angry Lion?

Special Requirements: We will start with a blintz bird base and there are some open sinks, squashes, swivels...

### Rabbit with Pointy Teeth

Level: Complex

Creator: Joel Garcia Moix

Teacher: Joel Garcia Moix

Description:
A Killer Rabbit of Caerbannog, a fearless creature attacking King Arthur in the movie "Monty Python and the Holy Grail."

Special Requirements: Pointy teeths should not scare you, even if they need small folds and sinks of all kinds-- open, closed and partially closed.

### Girl in Short Dress

Level: High Intermediate

Creator: Joel Garcia Moix

Teacher: Joel Garcia Moix

Description:
A cute girl, with a lot of personality, in a short dress

Special Requirements: This girl is made with multiple open sinks, so if you are familiar with them and can do some tiny folds, she will look great. 10-inch kami works well.

### Girl in Swimming Suit

Level: High Intermediate

Creator: Joel Garcia Moix

Teacher: Joel Garcia Moix

Description:

Special Requirements: We will transform a Bird base into a girl, with some open sinks and other magic folds. Best paper: Single colored, 21x21cm (8-inches) or bigger-- will be provided for the class.

### Robin

Level: High Intermediate

Creator: David Brill

Teacher: Jamie Kelley

Description:
David Brill’s 3D robin

Special Requirements: High precision is required to fold the model successfully. There are a couple of diabolical outside reverse folds included. Best paper is two-sided Kami.

### Resting Doe

Level: Low Intermediate

Creator: Joel Stern

Teacher: Joel Stern

Description:
This "minimalist" doe is made from two pieces that lock together -- no glue required.

Special Requirements: It's best made from sheets colored the same on both sides. 6-inch Tant, with its textures, is perfect for this and will be supplied for the class.

### Sonobe Module Workshop

Level: Intermediate

Creator: Mitsonobu Sonobe & others

Teacher: Michael Naughton

Description:
Mitsonobu Sonobè's module is an excellent one for beginners and experts alike. Literally dozens of models are possible and the workshop format allows each student to learn the one(s) they like. Good for those looking for an introduction to modular origami as well as those looking for more of a challenge. Models made from 1 to 60 modules will be shown. Construction ranges from low intermediate up to challenging. The basic module and some variations will be presented.

### 3 Easy Models

Level: Simple

Teacher: Kathy Hammell

Description:
This class focuses on some basic techniques for beginners. We will create models from index cards and from the water bomb base. There will be no cuts or glue; all models are made from one piece of paper. The following models will be included: Standing Heart and Hopping Frog from an index card and Butterfly from a water bomb base. Others may be taught as time permits.

### $Ice Cream Models Level: Low Intermediate Creator: Glenn Sapaden Teacher: Glenn Sapaden Description: Here are a number of models centered around my love for ice cream. We will fold at least two of the models (the soft serve cone and the Popsicle) and others if time permits. Special Requirements: Students should bring multiple crisp, foldable dollar bills to class. ###$ Pig

Level: High Intermediate

Description:
I teach this model to celebrate the Year of the Pig. In this version of Kade Chan’s model, we incorporate the dollar’s design to give the eyes to the pig. The model is deemed high intermediate due to the precise folding required and the utilization of the bill’s landmarks.

Special Requirements: Students should bring crisp, foldable dollar bills (preferably with even borders) to class.

### Simple Dog

Level: Intermediate

Creator: Enrique Martinez

Teacher: Enrique Martinez

Description:
This is a simple-looking dog which can be folded quite quickly once you know how to do it. If this class can be scheduled together with Oriol’s horse, it will become much more than just two models one after the other. Join us for the ride! Otherwise, join me for the ride! It will be my only 45 minute class.

Special Requirements: Having no fear of ugly folding and/or mistakes is expected. Please join this class if you already have achieved this level of uncaring or if you want to achieve it. We will fold for the sake of folding and won't care about the results, only about having a good time. We might learn something in the process.

### Baby Elephant

Level: High Intermediate

Creator: Enrique Martinez

Teacher: Enrique Martinez 1/4 OAS

Description:
A simple 3D baby elephant. Quite straightforward from a fish base. It can be wet folded.

Special Requirements: You must want to fold a baby elephant to take this class. Furthermore, you need to be fully committed to having some fun. Please leave worries at the door, specially the “I can’t fold well enough, I might fail” kind. If you fail, fail with gusto.

### Baby Wolfie Mija

Level: Complex

Creator: Enrique Martinez

Teacher: Enrique Martinez

Description:
Meant to be an origami version of a particularly sweet toy schnauzer. The bone in the picture was a present from her buried under my pillow. Rated complex because I honestly don’t know how to rate models. if you’re comfortable folding high intermediate, please feel free to join the class.

Special Requirements: A playful spirit and no fear of making a mistake are needed to follow this crazy dog. We will jump through some non-standard folds and you need to err on the side of throwing care into the wind and opening yourself up to failing miserably -- or the complete opposite. Take a chance and have fun.

### Penguin

Level: Complex

Creator: Enrique Martinez

Teacher: Enrique Martinez

Description:
3D shaped penguin with an open belly. This is a challenging but achievable model. Please try if you’re comfortable folding high intermediate models even though it is rated complex.

Special Requirements: You’ll need an icy glare and the determination to not stop if you don’t want the model to melt in your hands. That is a requirement for all my classes. Also you need to want to have fun. 10” two colored paper will also be helpful.

### Shirt Gift Bag

Level: Intermediate

Creator: Unknown

Teacher: Maureen Miller-Calamo

Description:
A bag shaped like a shirt will be folded from a square. If time permits, embellishments will be added (materials supplied by teacher). However, for filling the bag with a gift you will be on your own.

### Flower Parts

Level: Intermediate

Creator: Michie Takayama

Teacher: Kay Eng

Description:
This model is folded from a single square and results in a pretty 2D flat flower on a square base. If the corners are folded back it can be used to make a pin. The folding sequence is nice, with landmarks for each step.

### Pentagon Medal

Level: Low Intermediate

Creator: Takaya Iwamoto

Teacher: Takaya Iwamoto

Description:
You will learn how to fold this model using duo paper with a crease pattern printed on one side, enclosed in a circle. After the crease lines are scored, a circle will be cut out by a scissor.

Special Requirements: Bring a ruler for scoring, and a scissor for cutting out a circle. I will provide the specially prepared paper for the class.

### Square Medal

Level: Simple

Creator: Takaya Iwamoto

Teacher: Takaya Iwamoto

Description:
You will learn how to fold this model using duo paper with a crease pattern printed on one side,enclosed in a circle. After the crease lines are scored, a circle will be cut out by a scissor.

Special Requirements: Bring a ruler for scoring and a scissor for cutting out a circle. I will provide the specially prepared paper for the class.

### Triangular Medal

Level: Simple

Creator: Takaya Iwamoto

Teacher: Takaya Iwamoto

Description:
You will learn how to fold this model using duo papers with a crease pattern printed on one side, enclosed in a circle. After the crease lines are scored, a circle will be cut out by a scissor.

Special Requirements: Bring a ruler for scoring and a scissor for cutting out a circle. I will provide the specially-prepared paper for the class.

### Monika Star

Level: Low Intermediate

Creator: Maria Sinayskaya

Teacher: Maria Velazquez

Description:
This is a six-piece modular star

Special Requirements: 2.5 or 3" patterned paper

### Sheep 2019

Level: High Intermediate

Creator: Marc Vigo

Teacher: Marc Vigo

Description:
This sheep is my latest creation.

Special Requirements: There are no difficult folding steps required -- only some modeling for the head.

### Fairy

Level: Complex

Creator: Marc Vigo

Teacher: Marc Vigo

Description:
This fairy may be folded using the wet folding technique, although I usually fold with a dry sheet of tant paper. The folds are not complex but you have to interpret the instructions a little bit in order to achieve a cute fairy.

Special Requirements: Previous experience on wet/curved folding is highly recommended for the participants.

### Swallow

Level: High Intermediate

Creator: Marc Vigo

Teacher: Marc Vigo

Description:
This swallow is one of my latest designs, resulting in a 3D model with a color changed body that has volume.

Special Requirements: The hardest step of this model is a closed sink that forms a 3D belly. The rest of the steps are performed in 2D and are very geometric. No free modeling is required.

### YAHOOWL - Owl

Level: High Intermediate

Creator: Marc Vigo

Teacher: Marc Vigo

Description:
The YAHOOWL (Yet Another Humble Origami Owl) is a model that exploits the color change to achieve an expressive owl standing on its feet.

Special Requirements: Folding this model requires intermediate origami techniques. The hardest maneuvers are in and out reverse folding and a narrow crimp that locks the body and gives shape to it.

### Little Dragon

Level: High Intermediate

Creator: Marc Vigo

Teacher: Marc Vigo

Description:
Dragons are one of my favourite origami subjects. This one is my last creation from 2018. Instructions for folding a dragon with longer wings will be given at the end of the class.

Special Requirements: The model is founded on a 22.5 degree base. Mainly classical folds are required: petal, stretching, closed sink, etc. The hardest part is forming the tiny details of the head. Although 10'' kami is o.k., students may prefer a better or bigger paper. Color kraft is also o.k. Tissue foil or washi deluxe are perfect.

### Ninja Star

Level: Intermediate

Creator: Jose Meeusen

Teacher: Chila Caldera

Description:
You will learn how to fold this 8-unit modular with 6" squares of solid/solid duo (provided) and then you will fold another one using eight 4" squares of pattern/solid duo (provided). We should be able to make both models in the double-session, else I can help you finish later.

Special Requirements: This model involves making eight identical units, then assembling the finished model. The units are intermediate, and the assembly is not very difficult. No cutting or glue is needed. The 2nd model will be made with patterned paper so you can make a model that exhibits a pattern-in-a-pattern when finished. As for all modulars: careful, accurate work is required for the best result.

### Emmas Secret Star

Level: Intermediate

Creator: Wenhau Chao

Teacher: Chila Caldera

Description:
You will learn how to fold this star using one 6" square of duo paper (provided) and then you will fold another one using a square of 6" Kaleidoscope Duo (provided). We should be able to make both models in one session, else I can help you finish later.

Special Requirements: This model involves making an octagon from a square (in class), then precreasing the octagon, then collapsing the octagon, then shaping the points. You will need to bring or borrow scissors to remove the extraneous corners (or you can tear them off). Careful, accurate work is required for the best result. No glue is needed.

### Crocodile

Level: High Intermediate

Creator: Oriol Esteve

Teacher: Oriol Esteve

Description:
This is a cute crocodile that is inspired more by caricatures than the fierce wild animal itself. The distribution of colors works even on the belly side.

### Horse

Level: Intermediate

Creator: Oriol Esteve

Teacher: Oriol Esteve

Description:
This is an easy model with no complicated steps. The colorored mane comes from a wow effect maneuver.

### Unicorn

Level: High Intermediate

Creator: Oriol Esteve

Teacher: Oriol Esteve

Description:
This model has remarkable color changes and offers the opportunity to shape the mane to your taste.

### Mouse

Level: High Intermediate

Creator: Oriol Esteve

Teacher: Oriol Esteve

Description:
This cute model has tiny color changes on the head and nice 3D ears to achieve great expresiveness.

### Mr Cube

Level: High Intermediate

Teacher: Louise Oppenheimer-Flax

Description:
Box pleated person, folded from a 2 x 1 rectangular piece of paper divided into sevenths vertically. Materials will be provided for the class.

### Koneko (Kitten)

Level: Low Intermediate

Creator: Shunichi Ashimura

Teacher: Ayumi Hayatsu

Description:
This model is folded from 2 pieces of paper -- one for the head and one for the body.

### Toge-Toge Wreath

Level: Low Intermediate

Creator: Makoto Yamaguchi

Teacher: Ayumi Hayatsu

Description:
In this class we will assemble a 7-unit version of the model. It can also be assembled with 12 pieces and 30 pieces.

Special Requirements: Please bring in 3" paper (x 7).

### Crown Box

Level: Low Intermediate

Creator: Damien Marcotte

Teacher: Ron Arruda

Description:
Single sheet, open-topped box with pointed sides. 10-inch duo kami will be provided for the class.

### Modular Crown

Level: Low Intermediate

Creator: Laura Kruskal

Teacher: Ron Arruda

Description:
Modular crown made from 4 sheets of letter-size paper, which will be provided for the class.

### Kruskal Crown - Canoe Module

Level: Simple

Creator: Laura Kruskal

Teacher: Wendy Zeichner

Description:
This was one of Laura's Kruskal's favorite crowns, made from her famous Canoe Module. It is not diagrammed. A rainbow of paper colors will be provided in two sizes. To make a crown from the letter paper takes 10 modules. To make a crown from one-quarter of letter size takes 20-24 modules depending on your head size. Students can complete the letter-paper crown in class. The crown from smaller paper requires a bit of glue-- but it's worth it! No glue is needed for the Letter size. See photos!

### Rabbit

Level: High Intermediate

Creator: Hideo Komatsu

Teacher: Martin Quinn

Description:
For all those intermediate folders out there who would like to take the next step here is a gentle first step into the world of high intermediate models. Hideo Komatsu is well known not only for the beauty of his models, but also, and more importantly, for his meticulous attention to the folding sequence, ensuring that that in itself is a fun experience, being precisely specified with lots of landmarks.

Special Requirements: This model does include a pair of closed sinks, but these are well defined and are about as simple as closed sinks can get. In addition, in the latter stages of the model there will be folding steps after the model becomes 3D. Paper will be provided for the class.

### Eucalyptus Seedpod (Gum Nut)

Level: High Intermediate

Creator: Winnie Leung

Teacher: Winnie Leung

Description:
In this class, we will be folding a gum nut, a seedpod from gum trees (properly known as Eucalyptus), commonly found in Australia. I designed this model using the ORI-REVO software tool developed by Jun Mitani.

Special Requirements: Making this model involves curved folding techniques. The class will be provided with Elephant hide paper that is already scored.

### Plate (Two-Piece)

Level: Low Intermediate

Creator: Noriko Nagaat

Teacher: Patricia Grodner

Description:
Take 2 sheets of your favorite papers and transform them into a delightful bowl or help yourself to the paper that will be provided for the class.

### Hydrangea+Clover Tesselation

Level: High Intermediate

Creator: Shuzo Fujimoto

Teacher: Grant Goulart

Description:
The model that will be taught in this class is a composite of the clover and hydrangea tesselations. A crease pattern will be provided but the class will not be taught using it. The CP will be useful in understanding the model.

Special Requirements: An understanding of both the clover and hydrangea tesselations is important (both are designed by Shuzo Fujimoto). Students should also come with a square pre-folded into a 32x32 grid and every other diagonal. Students may find tweezers helpful.


text:
# Problem Description

Say you have an array for which the ith element is the price of a given stock on day i.

If you were only permitted to complete at most one transaction (ie, buy one and sell one share of the stock), design an algorithm to find the maximum profit.

# Analysis

## Idea 1

When I saw this problem, my first thought was to go through all of the prices and find the min and max in O(n) time and compute the maximum profit with (max - min).

However, this idea is wrong. Let’s see the following graph.

The min and max of this sotck price graph is min2 and max1. But (max1 - min2) is not what we want since we can’t sell a stock before buy it(max1 is before min2).

## Idea 2

Then I try to solve it using Dynamic Programming.

Let OPT(i, k) denotes the maximum we get at day k if we buy the stock at day i.

Let $P_i$ denotes the price of stock at day i.

Let optTal denotes the maximum we get.

Then we have:

$OPT(i, k) = \begin{cases} OPT(i, k - 1) & \text{not sell at day k} \\ P_k - P_i & \text{sell at day k} \end{cases}$

and the optimal:

\begin{align} optTotal = max\{OPT(i, j)\} \\ 1 \le i \le n \\ i \le j \le n \end{align}

With this formula, we can calculate the correct answer in $O(n^2)$ time.

However, this solution is not efficient and will exceed the time limit of LeetcodeOJ.

## Idea 3

We can go through all of the prices and update the min. Compute the profit we can get at day i by calculate ($P_i$ - min), update the max if we get a higher profit.

The complexity is O(n).

This solution works great!

format: plain


text: Proof. Would you like to be notified whenever we have a new post? Your email address will not be published. Law of Sines in "words": "The ratio of the sine of an angle in a triangle to the side opposite that angle is the same for each angle in the triangle." ], Adding $$h^2$$ to each side, $$a^2 + x^2 + h^2 = 2ax + y^2 + h^2$$, But from the two right triangles $$\triangle ACD$$ and $$\triangle ABD$$, $$x^2 + h^2 = b^2$$, and $$y^2 + h^2 = c^2$$. Here is a question from 2006 that was not archived: The Cut-the-Knot page includes several proofs, as does Wikipedia. Required fields are marked *. Applying the law of cosines we get So our equation becomes $$a^2 + b^2 = 2ax + c^2$$, Rearranging, we have our result: $$c^2 = a^2 + b^2 – 2ax$$. These equal ratios are called the Law of Sines. The definition of the dot product incorporates the law of cosines, so that the length of the vector from to is given by (7) (8) (9) where is the angle between and . This formula had better agree with the Pythagorean Theorem when = ∘. The formula can also be derived using a little geometry and simple algebra. When these angles are to be calculated, all three sides of the triangle should be known. LAW OF COSINES EQUATIONS They are: The proof will be for: This is based on the assumption that, if we can prove that equation, we can prove the other equations as well because the only difference is in the labeling of the points on the same triangle. If you never realized how much easier algebraic notation makes things, now you know! It can be used to derive the third side given two sides and the included angle. in pink, the areas a 2, b 2, and −2ab cos(γ) on the left and c 2 on the right; in blue, the triangle ABC twice, on the left, as well as on the right. Proof of the Law of Sines The Law of Sines states that for any triangle ABC, with sides a,b,c (see below) What I'm have trouble understanding is the way they define the triangle point A. Acute triangles. Theorem: The Law of Cosines To prove the theorem, we … Proof. No triangle can have two obtuse angles. In such cases, the law of cosines may be applied. The Law of Interactions: The whole is based on the parts and the interaction between them. Let side AM be h. In the right triangle ABM, the cosine of angle B is given by; And this theta is … Sin[A]/a = Sin[B]/b = Sin[C]/c. Two triangles ABD … Proof of the Law of Sines The Law of Sines states that for any triangle ABC, with sides a,b,c (see below) For more see Law of Sines. The wording “Law of Cosines” gets you thinking about the mechanics of the formula, not what it means. Direction Cosines. Sin[A]/a = Sin[B]/b = Sin[C]/c. or. Let side AM be h. Divide that number by 5, and you find that the angle of each triangle at the center of the pentagon is 72 degrees. We are a group of experienced volunteers whose main goal is to help you by answering your questions about math. PROOF OF LAW OF COSINES EQUATION CASE 1 All angles in the triangle are acute. Determine the measure of the angle at the center of the pentagon. The law of cosines calculator can help you solve a vast number of triangular problems. The law of cosine states that the square of any one side of a triangle is equal to the difference between the sum of squares of the other two sides and double the product of other sides and cosine angle included between them. Then, the lengths (angles) of the sides are given by the dot products: \cos(a) = \mathbf{u} \cdot \mathbf{v} Examples of General Formulas There are three versions of the cosine rule. It states that, if the length of two sides and the angle between them is known for a triangle, then we can determine the length of the third side. A picture of our triangle is shown below: Our triangle is triangle ABC. Law of cosines signifies the relation between the lengths of sides of a triangle with respect to the cosine of its angle. It is given by: Where a, b and c are the sides of a triangle and γ is the angle between a and b. We drop a perpendicular from point B to intersect with side AC at point D. That creates 2 right triangles (ABD and CBD). Since $$x = b\cos(C)$$, this is exactly the Law of Cosines, without explicit mention of cosines. In this case, let’s drop a perpendicular line from point A to point O on the side BC. Use the law of cosines to solve for a, because you can get the angle between those two congruent sides, plus you already know the length of the side opposite that angle. See Topic 16. To ask anything, just click here. A virtually identical proof is found in this page we also looked at last time: The next question was from a student who just guessed that there should be a way to modify the Pythagorean Theorem to work with non-right triangles; that is just what the Law of Cosines is. See the figure below. And so using the Laws of Sines and Cosines, we have completely solved the triangle. We can then use the definition of the sine of an angle of a right triangle. Calculate angles or sides of triangles with the Law of Cosines. In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi 's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Example 1: If α, β, and γ are the angles of a triangle, and a, b, and c are the lengths of the three sides opposite α, β, and γ, respectively, and a = 12, b = 7, and c = 6, then find the measure of β. We can use the Law of Cosines to find the length of a side or size of an angle. Proof. 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This splits the triangle into 2 right triangles. Referring to Figure 10, note that 1. This site uses Akismet to reduce spam. Law of cosines signifies the relation between the lengths of sides of a triangle with respect to the cosine of its angle. A proof of the law of cosines can be constructed as follows. In fact, we used the Pythagorean Theorem at least twice, first in the form of the distance formula, and again in the form of the Pythagorean identity, \sin^2 \theta + \cos^2 \theta = 1. So Law of Cosines tell us a squared is going to be b squared plus c squared, minus two times bc, times the cosine of theta. The law of cosine equation is useful for evaluating the third side of a triangle when the two other sides and their enclosed angle are known. The applet below illustrates a proof without words of the Law of Cosines that establishes a relationship between the angles and the side lengths of $$\Delta ABC$$: $$c^{2} = a^{2} + b^{2} - 2ab\cdot \mbox{cos}\gamma,$$ Proof of the law of sines: part 1 Draw an altitude of length h from vertex B. For a triangle with edges of length , and opposite angles of measure , and , respectively, the Law of Cosines states: In the case that one of the angles has measure (is a right angle), the corresponding statement reduces to the Pythagorean Theorem. Using notation as in Fig. Proof of the Law of Sines using altitudes Generally, there are several ways to prove the Law of Sines and the Law of Cosines, but I will provide one of each here: Let ABC be a triangle with angles A, B, C and sides a, b, c, such that angle A subtends side a, etc. Please provide your information below. Let u, v, and w denote the unit vector s from the center of the sphere to those corners of the triangle. As you drag the vertices (vectors) the magnitude of the cross product of the 2 vectors is updated. Here is my answer: The following are the formulas for cosine law for any triangles with sides a, b, c and angles A, B, C, respectively. In this mini-lesson, we will explore the world of the law of cosine. Now let us learn the law of cosines proof here; In the right triangle BCD, by the definition of cosine function: Subtracting above equation from side b, we get, In the triangle BCD, according to Sine definition, In the triangle ADB, if we apply the Pythagorean Theorem, then, Substituting for BD and DA from equations (1) and (2). As per the cosine law, if ABC is a triangle and α, β and γ are the angles between the sides the triangle respectively, then we have: The cosine law is used to determine the third side of a triangle when we know the lengths of the other two sides and the angle between them. From the above diagram, (10) (11) (12) Law of Cosines: Proof Without Words. Start with a scalene triangle ABC. in pink, the areas a 2, b 2, and −2ab cos(γ) on the left and c 2 on the right; in blue, the triangle ABC twice, on the left, as well as on the right. Law of cosines A proof of the law of cosines using Pythagorean Theorem and algebra. The main tool here is an identity already used in another proof of the Law of Cosines: Let a, b, c be the sides of the triangle and α, β, γ the angles opposite those sides. 2. Let's see how to use it. Ask Question Asked 5 months ago. Therefore, using the law of cosines, we can find the missing angle. Proof of the Law of Cosines Proof of the Law of Cosines The easiest way to prove this is by using the concepts of vector and dot product. The Law of Cosines is useful for finding: the third side of a triangle when we know two sides and the angle between them (like the example above) the angles of a triangle when we know all three sides (as in the following example) The equality of areas on the left and on the right gives . Viewed 260 times 10. The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines: + −  = where is the angle between sides and . First, here is a question we looked at last time asking about both the Law of Sines and the Law of Cosines; this time we’ll see the answer to the latter part: So the work is mostly algebra, with a trig identity thrown in. The Law of Cosines is also valid when the included angle is obtuse. Last week we looked at several proofs of the Law of Sines. $\vec a=\vec b-\vec c\,,$ and so we may calculate: The law of cosines formulated in this context states: 1. II. Draw an altitude of length h from vertex B. The Law of Cosines is presented as a geometric result that relates the parts of a triangle: While true, there’s a deeper principle at work. The proof depends on the Pythagorean Theorem, strangely enough! The Law of Cosines - Another PWW. Here we will see a couple proofs of the Law of Cosines; they are more or less equivalent, but take different perspectives – even one from before trigonometry and algebra were invented! Law of Cosines Law of Cosines: c 2 = a 2 + b 2 - 2abcosC The law of Cosines is a generalization of the Pythagorean Theorem. Calculates triangle perimeter, semi-perimeter, area, radius of inscribed circle, and radius of circumscribed circle around triangle. Ask Question Asked 5 months ago. In fact, we used the Pythagorean Theorem at least twice, first in the form of the distance formula, and again in the form of the Pythagorean identity, $$\sin^2 \theta + \cos^2 \theta = 1$$. The text surrounding the triangle gives a vector-based proof of the Law of Sines. From the cosine definition, we can express CE as a * cos(γ). Scroll down the page if you need more examples and solutions on how to use the Law of Cosines and how to proof the Law of Cosines. Let be embedded in a Cartesian coordinate systemby identifying: Thus by definition of sine and cosine: By the Distance Formula: Hence: Proof of the law of sines: part 1. If ABC is a triangle, then as per the statement of cosine law,  we have: – 2bc cos α, where a,b, and c are the sides of triangle and α is the angle between sides b and c. Fact: If any one of the angles, α, β or γ is equal to 90 degrees, then the above expression will justify the Pythagoras theorem, because cos 90 = 0. So I'm trying to understand a law of cosines proof that involves the distance formula and I'm having trouble. Law of Cosines: Proof Without Words. Proof of the Law of Cosines The Law of Cosines states that for any triangle ABC, with sides a,b,c For more see Law of Cosines. As a result, the Law of Cosines can be applied only if the following combinations are given: (1) Given two sides and the included angle, find a missing side. Law of cosine is not just restricted to right triangles, and it can be used for all types of triangles where we need to find any unknown side or unknown angle. If we label the triangle as in our previous figures, we have this: The theorem says, in the geometric language Euclid had to use, that: The square on the side opposite the acute angle [ $$c^2$$ ] is less than the sum of the squares on the sides containing the acute angle [ $$a^2 + b^2$$ ] by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls [a], and the straight line cut off within by the perpendicular towards the acute angle [x, so the rectangle is $$2ax$$]. The proof of the Law of Cosines requires that … Your email address will not be published. Let ABC be a triangle with sides a, b, c. We will show . The law of cosines is equivalent to the formula 1. PROOF OF LAW OF COSINES EQUATION CASE 1 All angles in the triangle are acute. The Law of Sines says that “given any triangle (not just a right angle triangle): if you divide the sine of any angle, by the length of the side opposite that angle, the result is the same regardless of which angle you choose”. Altitude h divides triangle ABC into right triangles AEB and CEB. Now the third angle you can simply find using angle sum property of triangle. As per the cosines law formula, to find the length of sides of triangle say △ABC, we can write as; And if we want to find the angles of △ABC, then the cosine rule is applied as; Where a, b and c are the lengths of sides of a triangle. Draw triangle ABC with sides a, b, and c, as above. Euclid has two propositions (one applying to an obtuse triangle, the other to acute), because negative numbers were not acceptable then (and the theorems don’t use numbers in the first place, but lengths!). In trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about the general triangles which relates the lengths of its sides to the cosine of one of its angles.Using notation as in Fig. This makes for a very interesting perspective on the proof! I won’t quote the proof, which uses different labels than mine; but putting it in algebraic terms, it amounts to this: From a previous theorem (Proposition II.7), $$a^2 + x^2 = 2ax + y^2$$, [This amounts to our algebraic fact that $$y^2 = (a – x)^2 = a^2 – 2ax + x^2$$. Law of cosines signifies the relation between the lengths of sides of a triangle with respect to the cosine of its angle. Another law of cosines proof that is relatively easy to understand uses Ptolemy's theorem: Assume we have the triangle ABC drawn in its circumcircle, as in the picture. Then BP = a-x. Required fields are marked *. Problem: A triangle ABC has sides a=10cm, b=7cm and c=5cm. In this article, I will be proving the law of cosines. First we need to find one angle using cosine law, say cos α = [b2 + c2 – a2]/2bc. Proof of the law of cosines The cosine rule can be proved by considering the case of a right triangle. These are not literally triangles (they can be called degenerate triangles), but the formula still works: it becomes mere addition or subtraction of lengths. FACTS to consider about Law of Cosines and triangles: 1. 1 $\begingroup$ I am trying to prove the Law of Cosines using the following diagram taken from Thomas' Calculus 11th edition. 1, the law of cosines states that: or, equivalently: Note that c is the side opposite of angle γ, and that a and b are the two sides enclosing γ. We have. Spherical Law of Cosines WewilldevelopaformulasimlartotheEuclideanLawofCosines.LetXYZ beatriangle,with anglesa,V,c andoppositesidelengthsa,b,c asshownintheﬁgure. cos(C) (the other two relationships can be proven similarly), draw an altitude h from angle B to side b, as shown below.. Altitude h divides triangle ABC into right triangles AEB and CEB. Your email address will not be published. The law of cosine equation is useful for evaluating the third side of a triangle when the two other sides and their enclosed angle are known. With that said, this is the law of cosines, and if you use the law of cosines, you could have done that problem we just did a lot faster because we just-- you know, you just have to set up the triangle and then just substitute into this, and you could have solved for a … The Law of Cosines is a theorem which relates the side-lengths and angles of a triangle.It can be derived in several different ways, the most common of which are listed in the "proofs" section below. cos(C) (the other two relationships can be proven similarly), draw an altitude h from angle B to side b, as shown below. $\Vert\vec a\Vert^2 = \Vert\vec b \Vert^2 + \Vert\vec c \Vert^2 - 2 \Vert \vec b\Vert\Vert\vec … This applet can help you visualize the aspects of one proof to the law of cosines. If ABC is a triangle, then as per the statement of cosine law, we have: a2 = b2 + c2 – 2bc cos α, where a,b, and c are the sides of triangle and α … Two triangles ABD and CBD are formed and they are both right triangles. But in that case, the cosine is negative. In a triangle, the sum of the measures of the interior angles is 180º. I've included the proof below from wikipedia that I'm trying to follow. It is also called the cosine rule. Proof of the Law of Cosines. So the Pythagorean Theorem can be seen as a special case of the Law of Cosines. Trigonometric proof using the law of cosines. CE equals FA. Now he gives an algebraic proof similar to the one above, but starting with geometry rather than coordinates, and avoiding trigonometry until the last step: (I’ve swapped the names of x and y from the original, to increase the similarity to our coordinate proof above.). Applying the Law of Cosines to each of the three angles, we have the three forms a^2 = b^2 … You will learn what is the law of cosines (also known as the cosine rule), the law of cosines formula, and its applications.Scroll down to find out when and how to use the law of cosines and check out the proofs of this law. Notice that the Law of Sines must work with at least two angles and two respective sides at a time. Hyperbolic case. It states that, if the length of two sides and the angle between them is known for a triangle, then we can determine the length of the third side. Construct the congruent triangle ADC, where AD = BC and DC = BA. Law of Cosines . For example, if all three sides of the triangle are known, the cosine rule allows one to find any of the angle measures. So I'm trying to understand a law of cosines proof that involves the distance formula and I'm having trouble. The law of cosines for the angles of a spherical triangle states that (16) (17) (18) You may find it interesting to see what happens when angle C is 0° or 180°! Call it D, the point where the altitude meets with line AC. Call it D, the point where the altitude meets with line AC. Drop a perpendicular from A to BC, meeting it at point P. Let the length AP be y, and the length CP be x. So, before reading the proof, you had better try to prove it. Since Triangle ABD and CBD … Active 5 months ago. 1$\begingroup$I am trying to prove the Law of Cosines using the following diagram taken from Thomas' Calculus 11th edition. 3. What is the Law of Cosines? When these angles are to be calculated, all three sides of the triangle should be known. The cosine rule can be proved by considering the case of a right triangle. First, use the Law of Cosines to solve a triangle if the length of the three sides is known. Proof of Law of Cosine Equation [Image will be Uploaded Soon] In the right triangle BAD, by the definition of cosine rule for angle : cos A = AD/c. Figure 7b cuts a hexagon in two different ways into smaller pieces, yielding a proof of the law of cosines in the case that the angle γ is obtuse. Now, find its angle ‘x’. We will try answering questions like what is meant by law of cosine, what are the general formulas of law of cosine, understand the law of cosine equation, derive law of cosine proof and discover other interesting aspects of it. The cosine rule, also known as the law of cosines, relates all 3 sides of a triangle with an angle of a triangle. 4. … It is also called the cosine rule. It is most useful for solving for missing information in a triangle. First, here is a question we looked at last time asking about both the Law of Sines and the Law of Cosines; this time we’ll see the answer to the latter part: Doctor Pete answered: So the work is mostly algebra, with a trig identity thrown in. In a triangle, the largest angle is opposite the longest side. Figure 7b cuts a hexagon in two different ways into smaller pieces, yielding a proof of the law of cosines in the case that the angle γ is obtuse. The heights from points B and D split the base AC by E and F, respectively. 1, the law of cosines states {\displaystyle c^ {2}=a^ {2}+b^ {2}-2ab\cos \gamma,} Law of Cosines. It is given by: First we need to find one angle using cosine law, say cos α = [b, Then we will find the second angle again using the same law, cos β = [a. In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle. It is important to solve more problems based on cosines law formula by changing the values of sides a, b & c and cross-check law of cosines calculator given above. 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text:
# Late-time tails of a Yang–Mills field on Minkowski and Schwarzschild backgrounds

```@article{Bizon2007LatetimeTO,
title={Late-time tails of a Yang–Mills field on Minkowski and Schwarzschild backgrounds},
author={Piotr Bizo'n and Tadeusz Chmaj and Andrzej Rostworowski},
journal={Classical and Quantum Gravity},
year={2007},
volume={24},
pages={F55 - F63}
}```
• Published 7 April 2007
• Mathematics
• Classical and Quantum Gravity
We study the late-time behaviour of spherically symmetric solutions of the Yang–Mills equations on Minkowski and Schwarzschild backgrounds. Using nonlinear perturbation theory we show in both cases that solutions having smooth compactly supported initial data possess tails which decay as t−4 at timelike infinity. Moreover, for small initial data on Minkowski background we derive the third-order formula for the amplitude of the tail and numerically confirm its accuracy.

## Figures from this paper

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• 2016
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• 2013
We show how matter can be included in a constrained ADM-like formulation of the Einstein equations on constant mean curvature surfaces. Previous results on the regularity of the equations at future
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• 2014
We consider a spherically symmetric (purely magnetic) SU(2) Yang-Mills field propagating on an ultrastatic spacetime with two asymptotically hyperbolic regions connected by a throat of radius
We study linear gravitational perturbations of Schwarzschild spacetime by solving numerically Regge–Wheeler–Zerilli equations in a time domain using hyperboloidal surfaces and a compactifying radial
• Physics
• 2008
We study gravitational perturbations of Schwarzschild spacetime by solving a hyperboloidal initial value problem for the Bardeen–Press equation. Compactification along hyperboloidal surfaces in a
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• 2013
We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving
• Physics
• 2018
We study critical behavior in gravitational collapse of a general spherically symmetric Yang-Mills field coupled to the Einstein equations. Unlike the magnetic ansatz used in previous numerical work,
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• 2016
Black holes (BHs) in general relativity (GR) are very simple objects. This property, that goes under the name of ‘no-hair’, has been refined in the last few decades and admits several versions. The
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• 2009
We study numerically the late-time behaviour of the coupled Einstein–Yang–Mills system. We restrict ourselves to spherical symmetry and employ Bondi-like coordinates with radial compactification.

## References

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• 1999
We investigate the late-time evolution of theYang-Mills field in the self-gravitating backgrounds:Schwarzschild and Reissner-Nordstrom spacetimes. Thelate-time power-law tails develop in the
• Mathematics
• 1982
In this paper and its sequel we shall prove the local and then the global existence of solutions of the classical Yang-Mills-Higgs equations in the temporal gauge. This paper proves local existence
In this paper I will first derive, based on energy estimations and geometric invariance, the asymptotic behavior of solutions of linear spin field equations in Minkowski space. It generalizes the
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Physical review. D, Particles and fields
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The branch-cut integral produces a weak late-time radiative power-law decay tail that will characterize the astrophysically observed radiation spectrum for times subsequent to the exponential decay of the quasinormal ringing.
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Yang-Mills equations in R3+1 is well-posed in the energy norm. This means that for an appropriate gauge condition, we construct local, unique solutions in a time interval which depends only on the
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We prove global solvability of the Cauchy problem for the Yang-Mills equations on smooth globally hyperbolic four dimensional Lorentzian manifolds.
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A general definition of symmetries of gauge fields is proposed and a method developed for constructing symmetric fields for an arbitrary gauge group. Scalar fields occur naturally in the formalism
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The Yang-Mills fields considered by us in an earlier paper are asymptotically non-interacting. Also any free field is an incoming field for some Yang-Mills field.

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text:
## Monday, August 12, 2013

### We don't know much about what the exchange rate ought to be

Many people are confidently saying that the market's price of the rupee-dollar exchange rate is wrong. They think they know what the exchange rate should be. There are many pitfalls along this path.

#### The simplest international trade perspective

To fix intuition, let's think there are only two countries: India and the US. Suppose there was no capital account. Suppose there was only import and export of goods and services on the current account. In the absence of a capital account, in every time period, the current account would have to work out to zero. The currency market would clear to yield an appropriate exchange rate from the viewpoint of export competitiveness. We would find an exchange rate at which we earn enough export proceeds to be able to pay for for our imports (that are also a function of this exchange rate).

Suppose we had a base year in which things were square. Now, from that point onwards, one could look at inflation in the two countries and the exchange rate and get a sense about how export competitiveness was changing. As an example, inflation in India is 10% and inflation in the US is 2% so we might think that the exchange rate should depreciate steadily at 8% per year in order to keep the CAD at 0.

The Real Effective Exchange Rate (REER) goes one step further and implements this calculation while taking into account the fact that India trades with many countries. A note of caution: The best REER measure for India is that made by the BIS.

#### Problem 1: The export basket is different from the CPI

The CPI is, indeed, the best measure of inflation. But the things that we export and import are not the things that the average household consumes. Hence, this crude estimate -- that the rupee should depreciate by roughly 8 per cent per year on average -- is wrong.

There is really no way to solve this problem. In an ideal world, we would have price indexes that are specific to the import and export basket. But these indexes are essentially impossible to make. Think of all the goods and services that India imports or exports, and the problems of obtaining a sensible price for each of them.

#### Problem 2: Productivity is not constant

All this assumes that nothing else is changing. But productivity is changing, slowly in the US and dramatically in India. In our backyard, there are a million microeconomic mutinies, through which production is becoming more efficient. Individuals learn. Firms improve their processes. Indian firms get internationalised, and global firms start operating here, all of which drives a ferment of improving knowledge. Our indirect tax policy & administration has slowly gotten better. State of the art equipment gets brought in. We learn how to utilise better and more specialised raw materials (which are often imported). Infrastructure gets better, thus linking up productive capacity in difficult locations (e.g. 100 km away from Bombay, 1000 km away from Bombay, 4000 km away from Bombay) to global markets.

The pair of graphs here is blindingly obvious and yet revealing. From 1994 till 2010, the Indian REER fluctuated between 85 and 110. Over this period, merchandise exports grew from \$2 billion a month to \$15 billion a month. What was going on? Superior Indian productivity growth.

Qingyuan Du, Shang-Jin Wei and Peichu Xie have a fascinating new paper Roads and the real exchange rate, which finds that changes in transport infrastructure have important implications for the real exchange rate.

#### Problem 3: The exchange rate is actually mostly about finance and not trade

So far, we have played in the simplest world where there was no capital account, and all that was going on was imports and exports (of goods and services). That's not the world that we live in anymore. We have a mostly open capital account, which gives each country the convenience of not having to achieve a CAD of 0 in every time period. We obtain microeconomic gains from international capital flows.

In 2012-13, India had inflows on the current account of \$452 billion and outflows of \$583 billion. If this was all that was going on, on the currency market, we would have trading volume of \$583 billion per year. The outflows would require buying \$583 billion. On the other side would be exporters selling \$452 billion and capital inflows filling the gap. This translates to (one way) turnover on the currency market of \$2.3 billion per day. In truth, the currency market for the rupee trades between \$40 billion and \$70 billion every day. The overall activity on the currency market dwarfs the minor business of sorting out the current account. The currency is now a financial product.

#### Bottom line: Exchange rate assessment is a mugs game

We started out with a simple and intuitive world, but ran into three problems. We can't quite use CPI in India and in trading partners, as the CPI is the basket consumed by the household and not a reflection of prices of the goods and services that are traded. The big story is all about productivity growth in India, which we know little about. And, the exchange rate is a financial object and not driven in the short run or medium term by trade considerations.

#### Policy makers should be cautious when thinking there is something going wrong with the market's exchange rate

When so little is known about exchange rate assessment, we should be cautious before determining that something is wrong with the exchange rate that the currency market has made.

None of this is new (1997, 2007). But every few years, we seem to come up against a situation where policy makers have strong views on what the exchange rate ought to be. We may all have our own personal opinions about what the exchange rate ought to be. A Hippocratic oath should prevent us from undertaking interventions in the real world that have manifestly visible costs [link, link], until we are really sure that there are corresponding gains.

1. Ajay, thanks for a great article. Is it that policy makers are guessing what the right exchange rate should be (a frightening prospect), or is it that they need to smoothen the transition to whatever the right market-determined rate is. The latter seems more like the concern here, because sharp volatility can create long term consequences, without time to adjust to the change...

2. While not directly relating to the subject of 'measuring an exchange rate', i see INR again weaker after the FM announces measures that includes issuance of quasi-sovereign bonds to be raised by PSU FinCos plus some likely relaxation on the ECBs.

Will it not be a better idea to pay, say 4% for 3 years, for all FCNR deposits - any PSU that would raise a 3-5 year bond will end up paying 300-350 basis over midswap.. Rather than adding that way to the external debt (when RBI already has a 280b reserves), it would be more useful if the NRI gets that benefit. A lock-in period or some attractive frills can be embedded so that the NRI sees value. An NRI will also close loans taken for the purpose of remittances - out of his earnings - which will help a non-disruptive process of closing out the liability (opposed to a PSU loan where halfyearly repayments will create sufficient noise in the onshore markets).

A run-down in gold reserves by say 5b will also help but it may have other ramifications internationally..

Why not the government invite prize-winning suggestions on how to tackle this currency crisis? Rather than leaving it to the panelists in prime channels (who range from agricultural ministers to film-personalities turned MPs to ever-pessimistic opposition leader, and as if this is not enough, a terrible moderator among them from the channel), it would be better to hold intelligent debates and discussions that pave a way for managing this crisis.

The worst is yet not over - a possible downgrade by major rating agencies (say the foodbill passage and its impact on the fiscals) is likely to make matters worse for the PSUs or whoever is likely to raise FC debt.

1. "Why not the government invite prize-winning suggestions on how to tackle this currency crisis?"

Did you even read the article? The admin should stop focusing on just relieving the symptom and focus on the real problem. The currency will be strong only if the economy is strong. Period. There is nothing else to be done to manage the volatility or direction or whatever. Focus on the fundamentals of the economy.

Secondly, what needs to be done is common sense. No need for prize winning suggestions. Its not rocket science.

2. One more point. Lets talk about how we got into the currency crisis situation and make sure we don't do that again. In other words, while we are all talking about how to fix this currency crisis, why don't we focus on the actual problem of how we got here? Why is the media and intelligentsia not asking the govt about this? Why is the govt allowed to behave like this currency crisis is some exogenous phenomenon that occurred independent of its disastrous policies. Lets talk about how we got here, shall we?

3. Agreed, Mr/Miss/Mrs Anonymous.
Do you have a suggestion or suggestions to offer?
i don't have any, honestly.
There is none actually that will work wonders overnight.
Its becoming a cliche to hear 'focus on fundamentals and don't treat the symptoms'. If it was so clear at a macro level, someone as distinguished as Subbarao would have done differently from whatever we have seen in the last 3 months..
Govt is afterall the same set of people that was earlier common ma(e)n Manmohan Singh is not someone who does not understand Central Banking nor is he new to currency crisis. And, if someone had known 'we will get into this currency crisis situation', probably he or she would not have pushed it either. I don't know but if it just common sense, we wouldnt be battling for weeks at a stretch..

4. Huh? You seem to have missed the point and the article.

What specific suggestion are you looking for? For the Droopee? Nothing needs to be done. What exactly are you hoping for? Didn't you read the article? For the CAD, again, nothing needs to be done (currency depreciation helps with that to some extent).

And, we need to get out of this habit of short-termism and focus on fundamentals of domestic production and export sector, to make them more competitive by sound long term policies.

And, you can just see what other non-establishment economists have been saying for the past 2-3 years to see that it is indeed commonsense as to what needs to be done. The threat of CAD crisis, capital flight, currency, inflation are not new problems. Just read what other economists have been saying in any of the newspapers, for quite some time now.

Just because something is a cliche, doesn't make it wrong. The commonsense question is why have these cliched focus on fundamentals not been carried out for such a long period? Why do we expect or deserve any better when the fundamentals have not been looked after for so long?

I find your comments on RBI and MMS amusing. Do you have any evidence of their expertise? The very fact that a) they are battling this for weeks, and b) in the most childish manner possible, is evidence to the contrary.

Actually, there is an overnight fix for everything and it is coming up:

Raghuram Rajan’s handsomeness averts India’s economic crisis

5. I agree - non-establishment economists have indeed been warning. To a very large extent even the outgoing RBI Governor has been warning (and it requires courage to do so in the Indian bureaucratic setup). Not holding his brief just because of intimacy, but i would give him all credit for whatever has been done.

i started my post with a caveat that my comments may not directly relate to the subject article so please understand i have read this wonderful piece and am in agreement. like many in INDIA, i am a great fan of Ajay Shah.

i started a tangentially different string with a comment on the futility of these short term measures - which is what you also endorse.

I know i cannot say US has a long history of CAD. USD enjoys the reserve currency status one might say and hence capital flows compensate. Australia has no less a different history. Current Account to GDP in Australia averaged -4.20 pct for 30 years from 1980, reaching record low of -6.20 pct in December of 2007. But that would also go down as a different league game. Unfortunately we are in the league of Turkey and South Africa.

This argument can go on endlessly - and that is not my intention - i am not an economist - just a common (sensed) man..

In many ways Bernanke and Subba Rao rode the roughest times of their respective economies - Let us hope the Rajans and Summers or Yellens of the world get the luck of the Carneys of the world..

MMS's credentials are i thought on public domain - his background as an Oxford economist, his positions in UNCTAD, finance ministry, planning commission, his stellar role in early 1990s when the country had barely half a months fx reserves.. what more do we need. Maybe his folly was to abolish the licence raj at the instance of IMF - but that was a tradeoff for dough in the bowl we were holding.. Times are different and monetary policies have not been effective.. Central banks are growingly losing their independence and a total obfuscation will be the way forward

Will be extremely happy if an overnight fix is coming and that is successful..

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