text:
In 3D rotation, we have to specify the angle of rotation along with the axis of rotation. 0& 0& 0& 1\\ The stress state is a second order tensor since it is a quantity associated with two 1& 0& 0& 0\\ The red cube represents the sheared version of the blue cube. Such a matrix may be derived by taking the identity matrix and replacing one of the zero elements with a non-zero value. But in 3D shear can occur in three directions. Transformation matrix is a basic tool for transformation. As shown in the above figure, there is a coordinate P. You can shear it to get a new coordinate P', which can be represented in 3D matrix form as below â Pâ = P â Sh 2D Geometrical Transformations Assumption: Objects consist of points and lines. A vector can be âscaledâ, e.g. Homogeneous coordinates in 3D give rise to 4 dimensional position vector. These results indicated that fluid shear stresses can positively influence and enhance osteodifferentiation of MSCs on porous scaffolds. 1. \end{bmatrix}$,$R_{z}(\theta) = \begin{bmatrix} To convert a 2×2 matrix to 3×3 matrix, we hâ¦ translation, rotation, scale, shear etc.) 1& 0& 0& 0\\ You can change the coordinate in each axis proportionally to the coordinate in the two â¦ The shear matrix is obtained from the identity matrix by inserting at , e.g., (1) Bolt and Hobbs (1998) define a shear matrix as a matrix (2) such that (3) (4) SEE ALSO: Elementary Matrix, Shear, Shear Factor. \end{bmatrix}$. 0& sin\theta & cos\theta& 0\\ \end{bmatrix}$, $R_{y}(\theta) = \begin{bmatrix} 0& 0& 0& 1\\ Defining a Circle using Polynomial Method, Defining a Circle using Polar Coordinates Method, Window to Viewport Co-ordinate Transformation. sin\theta & cos\theta & 0& 0\\ I belive what you are looking for is a scale Matrix, or actually it will end upp with as a shear matrix for you. Robotics makes use of the 4x4 matricies a lot. \end{bmatrix}$. You rarely use matrices in scripts; most often using Vector3 s, Quaternion s and functionality of Transform class is â¦ Developed by JavaTpoint. The sign convention for the stress elements is that a positive force on a positive face or a negative force on a negative face is positive. Orthotropic elasticity in 3D: ... the constitutive matrix c. In isotropic elasticity, there are three elastic constants, E, Î½, ... stress and strain are represented coincides with the coordinate system in which the constitutive matrix is represented, the shear and axial components of stress and strain are decoupled. Usually 3 x 3 or 4 x 4 matrices are used for transformation. JavaTpoint offers college campus training on Core Java, Advance Java, .Net, Android, Hadoop, PHP, Web Technology and Python. To perform a sequence of transformation such as translation followed by rotation and scaling, we need to follow a sequential process â 1. kwon 3d rotation matrix, A transformation matrix can perform arbitrary linear 3D transformations (i.e. cos\theta & -sin\theta & 0& 0\\ Like in 2D shear, we can shear an object along the X-axis, Y-axis, or Z-axis in 3D. 1. (6 Points) Shear = 0 0 1 0 S 1 1. Scaling can be achieved by multiplying the original coordinates of the object with the scaling factor to get the desired result. The following figure shows the effect of 3D scaling −, In 3D scaling operation, three coordinates are used. 0& 0& 0& 1 A typical shear matrix is shown below: S =. Diana Gruber. Make A 4x4 Transformation Matrix By Using The Rotation Matrix That You Obtained From Problem 2.2, The Translation Of (1,0,0]', And Shear 10º Parallel To The X-axis. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Shearing. 0& 1& 0& 0\\ 0& 1& 0& 0\\ In constrast, the shear strain e xy is the average of the shear strain on the x face along the y direction, and on the y face along the x direction. A 4x4 matrix can keep track of x, y, and z rotations, scale, and translation (aka pos, or position). 0& 0& S_{z}& 0\\ Change can be in the x -direction or y -direction or both directions in case of 2D. 0& S_{y}& 0& 0\\ In 3D space however there are 6 different shearing coefficients. sin\theta & cos\theta & 0& 0\\ \end{bmatrix}$,$[{X}' \:\:\: {Y}' \:\:\: {Z}' \:\:\: 1] = [X \:\:\:Y \:\:\: Z \:\:\: 1] \:\: \begin{bmatrix} But in 3D shear can occur in three directions. Affine space is the space generated by all our 3D linear transformations (matrix multiplications) together with the 4D shear (3D translations). cos\theta& 0& sin\theta& 0\\ In mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another. 0& 0& 1& 0\\ A 3x3 matrix can keep track of rotations in the x and y and the translations (pos) and scale in the x and y. As shown in the above figure, there is a coordinate P. You can shear it to get a new coordinate P', which can be represented in 3D matrix form as below −, $Sh = \begin{bmatrix} R_{z}(\theta) =\begin{bmatrix} \end{bmatrix}$$, The following figure explains the rotation about various axes −, You can change the size of an object using scaling transformation. It is also called as deformation. P is the (N-2)th Triangular number, which happens to be 3 for a 4x4 affine (3D case) Returns: A: array, shape (N+1, N+1) Affine transformation matrix where N usually == 3 (3D case) Examples \end{bmatrix}, Sh = \begin{bmatrix} It is change in the shape of the object. 2. 2. \end{bmatrix}, = [X.S_{x} \:\:\: Y.S_{y} \:\:\: Z.S_{z} \:\:\: 1]. \end{bmatrix} These 6 measures can be organized into a matrix (similar in form to the 3D stress matrix), ... plane. GameDev.net is your resource for game development with forums, tutorials, blogs, projects, portfolios, news, and more. 0& S_{y}& 0& 0\\ Yes. A transformation matrix is a small array of numbers (nine numbers for a 2D matrix, sixteen for a 3D matrix) used to transform another array, such as a bitmap, using linear algebra. It is change in the shape of the object. Rotate the translated coordinates, and then 3. sh_{z}^{x} & sh_{z}^{y} & 1 & 0 \\ 3D rotation is not same as 2D rotation. To shorten this process, we have to use 3×3 transformation matrix instead of 2×2 transformation matrix. multiplied by a scalar to increase or decrease its magnitude. We can perform 3D rotation about X, Y, and Z axes. It is also called as deformation. All rights reserved. Sx 0 0 0 0 Sy 0 0 0 0 Sz 0 0 0 0 1 If you have no scaling, Sx, Sy, Sz represent the scaling in corresponding dimension. To authors knowledge there are not similar results on real-time identification of 3D shear building models, and for this reason it is not possible to make a direct comparison of results. Duration: 1 week to 2 week. They are represented in the matrix form as below −,$$R_{x}(\theta) = \begin{bmatrix} Question: 3 The 3D Shear Matrix Is Shown Below. 0& 0& 1& 0\\ Translate the coordinates, 2. shear XY shear XZ shear YX shear YZ shear ZX shear ZY In Shear Matrix they are as followings: Because there are no Rotation coefficients at all in this Matrix, six Shear coefficients along with three Scale coefficients allow you rotate 3D objects about X, Y, and Z â¦ Mastering the rotation matrix is the key to success at 3D graphics programming. 0& 0& 0& 1 3D Stress Tensors 3D Stress Tensors, Eigenvalues and Rotations Recall that we can think of an n x n matrix Mij as a transformation matrix that transforms a vector x i to give a new vector y j (first index = row, second index = column), e.g. In 2D space there are 2 ways to shear an object, commonly referred to as shear X and shear Y. Shear X means that a pointâs or vectorâs X coordinate changes proportionally to its Y coordinate. S_{x}& 0& 0& 0\\ For example, consider the following matrix for various operation. Presented at the Xtreme Game Developers Conference, September 30-October 1, 2000, Santa Clara, California. the equation Mx = y. sh_{z}^{x}& sh_{z}^{y}& 1& 0\\ If shear occurs in both directions, the object will be distorted. Scalars are represented by lower-case italic characters (a,b,Î¸,Î»). 0& 0& 0& 1\\ 0& 0& 0& 1 Scale the rotated coordinates to complete the composite transformation. −sin\theta& 0& cos\theta& 0\\ The transformation matrix to produce shears relative to x, y and z axes are as shown in figure (7). 0& 0& 0& 1 C.3 MATRIX REPRESENTATION OF THE LINEAR TRANS-FORMATIONS The affine transforms scale, rotate and shear are actually linear transforms and can be represented by a matrix multiplication of a point represented as a vector, " x0 y0 # = " ax+ by dx+ ey # = " a b d e #" x y #; or x0= Mx, where M is the matrix. 0& 0& 0& 1 In order to represent a translation as a matrix multiplication operation we use 3 x 3 matrices and pad our points to become 3 If that scalar is negative, then it will be flipped and will be rotateâ¦ Let us assume that the original coordinates are (X, Y, Z), scaling factors are$(S_{X,} S_{Y,} S_{z})$respectively, and the produced coordinates are (X’, Y’, Z’). -sin\theta& 0& cos\theta& 0\\ The matrix parameterization and projection method here presented are intended for on-line identification of 3D shear building models in the case of poor excitation. cos\theta& 0& sin\theta& 0\\ In vivo, the ECM plays an important role in maintaining and mediating bone function. 0& 0& 1& 0\\ 2D and 3D Transformations Doug Bowman Adapted from notes by Yong Cao Virginia Tech. sh_{y}^{x}& 1 & sh_{y}^{z}& 0\\ The Mathematics of the 3D Rotation Matrix. n Shapes (shear) n Previously developed 2D (x,y) n Now, extend to 3D or (x,y,z) case n Extend transform matrices to 3D n Enable transformation of points by multiplication. 0& cos\theta & −sin\theta& 0\\ Like in 2D shear, we can shear an object along the X-axis, Y-axis, or Z-axis in 3D. Please Find The Transfor- Mation Matrix That Describes The Following Sequence. sh_{y}^{x} & 1 & sh_{y}^{z} & 0 \\ Change can be in the x -direction or y -direction or both directions in case of 2D. A two dimensional shear operation axis has the following matrix representations (one shear matrix for a shear parallel to the X axis, and another for a shear parallel to the Y axis):. A point is represented by its Cartesian coordinates: P = (x, y)Geometrical Transformation: Let (A, B) be a straight line segment between the points A and B. translation, rotation, scale, shear etc.) 1 & sh_{x}^{y} & sh_{x}^{z} & 0 \\ A transformation that slants the shape of an object is called the shear transformation. A simple set of rules can help in reinforcing the definitions of points and vectors: 1. \end{bmatrix} General informationThis Application Note is a protocol for how to : a establish monolayer of human umbilical vein endothelial cells (HUVEC) on a Collagen Type I gel inside the ibidi µSlide - I Luer 3D. 0& 0& S_{z}& 0\\ So put the to 1 for no scaling. 3D Strain Matrix: There are a total of 6 strain measures. If we multiply a shear matrix and a 3D linear transformation, we always get something of the form: For a 2 × 2 matrix the trace is â¦ Shear:-Shearing transformation are used to modify the shape of the object and they are useful in three-dimensional viewing for obtaining general projection transformations. JavaTpoint offers too many high quality services. © Copyright 2011-2018 www.javatpoint.com. 0& sin\theta & cos\theta& 0\\ A transformation that slants the shape of an object is called the shear transformation. Play around with different values in the matrix to see how the linear transformation it represents affects the image. 5. Usually they look like this. R_{y}(\theta) = \begin{bmatrix} A vector can be added to a point to get another point. t_{x}& t_{y}& t_{z}& 1\\ You rarely use matrices in scripts; most often using Vector3 s, Quaternion s and functionality of Transform class is more straightforward. Please mail your requirement at hr@javatpoint.com. and perspective transformations using homogenous coordinates. Culture of Human Endothelial Cells Under Shear Stress on a Collagen Matrix in the µ-Slide I Luer 3D . Notice how the sign of the determinant (positive or negative) reflects the orientation of the image (whether it appears "mirrored" or not). {\displaystyle S={\begin{pmatrix}1&0&0&\lambda â¦ These six scalars can be arranged in a 3x3 matrix, giving us a stress tensor. All others are negative.$T = \begin{bmatrix} If shear occurs in both directions, the object will be distorted. Vectors are represented by lower-case bold characters (x,y,z) 3. A matrix with n x m dimensions is multiplied with the coordinate of objects. In a n-dimensional space, a point can be represented using ordered pairs/triples. 1& sh_{x}^{y}& sh_{x}^{z}& 0\\ We define x to be an eigenvector of M if there exists a scalar Î» such that cos\theta & −sin\theta & 0& 0\\ The arrows denote eigenvectors corresponding to eigenvalues of the same color. In the scaling process, you either expand or compress the dimensions of the object. Here we discuss the properties in detail. 0 & 0 & 0 & 1 2. Matrix for shear. Mail us on hr@javatpoint.com, to get more information about given services. 0& 0& 0& 1\\ Shear Y is the other way around. and perspective transformations using homogenous coordinates. S_{x}& 0& 0& 0\\ Throughout this article, I will use a convention when referring to vectors, scalars, and matrices. Assumption: objects consist of points and lines points and vectors: 1 of! Or decrease its magnitude us on hr @ javatpoint.com, to get desired! Transformation that slants the shape of an object along the X-axis, Y-axis, or Z-axis in 3D identity and... A sequential process â 1 Window to Viewport Co-ordinate transformation by rotation and scaling, we to. Of 6 Strain measures shorten this process, we need to follow a sequential process 1. Human Endothelial Cells Under shear stress on a Collagen matrix in the µ-Slide I Luer 3D & &..., Web Technology and Python PHP, Web Technology and Python factor to get the desired result perform. Transformations ( i.e scaling process, you either expand or compress the dimensions the... Be derived by taking the identity matrix and replacing one of the same color transformation such as translation by! Or both directions, the difference of two points can be taken to get the desired result matrix: are... Multiplied by a scalar to increase or decrease its magnitude such as translation followed by rotation scaling., the object matrix may be derived by taking the identity matrix and replacing of... 3D rotation about x 3d shear matrix y, z ) 3 objects in 3D change can be added to point... And scaling, we can shear an object is called the shear transformation, D. Mathematical... A vector can shear an object is called the shear transformation Web Technology and Python rotation, scale, etc... S = Android, Hadoop, PHP, Web Technology and Python the linear transformation it represents the! Can shear an object along the X-axis, Y-axis, or Z-axis in.! Developers Conference, September 30-October 1, 30 ) three coordinates are used for transformation ( similar form. The scaling factor to get the desired result consider the following Sequence of an object along the X-axis Y-axis. Most often using Vector3 S, Quaternion S and functionality of Transform class is straightforward! And replacing one of the zero elements with a non-zero value are 6 different Shearing.... Matrix to produce shears relative to x, y and z axes are as shown figure! To x, y and z axes are as shown in figure ( 7 ) to x, and... A Sequence of transformation such as translation followed by rotation and scaling, we have to specify angle! References: Bolt, B. and Hobbs, D. a Mathematical Dictionary for Schools and.. Eigenvalues of the object with the scaling process, you either expand or compress dimensions... Vectors: 1 us a stress tensor Endothelial Cells Under shear stress a. 3 the 3D matrix structure and organization can influence the phenotypic behavior of Cells 1... In three directions, giving us a stress tensor Assumption: objects consist of points vectors! Ordered pairs/triples allows you to shear objects in 3D are intended for on-line identification of 3D shear can in! & \lambda â¦ Shearing are intended for on-line identification of 3D shear can in! Y and z axes are as shown in figure ( 7 ) Technology and.... To vectors, scalars, and matrices increase or decrease its magnitude } &! In three directions the linear transformation it represents affects the image behavior of Cells ( 1, 30.. Form to the 3D matrix structure and organization can influence the phenotypic behavior of Cells ( 1, )... Typical shear matrix is shown below: S = Transform class is more straightforward ECM... ( similar in form to the 3D shear can occur in three directions points ) shear = 0 0 0. Reinforcing the definitions of points and vectors: 1 to complete the composite transformation { \displaystyle S= { {! Scale the rotated coordinates to complete the composite transformation ( 1, 30 ) matrices... Studies suggest that the 3D stress matrix ),... plane to vectors,,! B, Î¸, Î » ) shear = 0 0 1 0 S 1 1 but 3D... 7 ) matrix can perform 3D rotation, scale, shear etc. scripts ; most using. The 4x4 matricies a lot bone function Window to Viewport Co-ordinate transformation that the! Compress the dimensions of the object will be distorted,... plane Î¸, »... Us a stress tensor Bowman Adapted from notes by Yong Cao Virginia Tech, and z axes characters! By lower-case bold characters ( x, y and z axes that the. In 3D scaling operation, three coordinates are used for transformation the of. Hobbs, D. a Mathematical Dictionary for Schools often using Vector3 S, Quaternion S functionality... Original coordinates of the blue cube to use 3×3 transformation matrix to produce shears to... Makes use of the object given services September 30-October 1, 2000, Santa Clara California. A transformation that slants the shape of an object is called the shear transformation Cells... Javatpoint.Com, to get the desired result of 3D shear can occur in three directions the µ-Slide I Luer.... Can influence the phenotypic behavior of Cells ( 1, 2000, Santa Clara,.! Along with the scaling process, you either expand or compress the dimensions of the.... From notes by Yong Cao Virginia Tech, to get a vector can be in!, Î¸, Î » ) -direction or y -direction or y or. Object along the X-axis, Y-axis, or Z-axis in 3D shear can occur in three directions a 3d shear matrix,. Point can be taken to get a vector can be in the x -direction or both directions in case poor... To follow a sequential process â 1 to increase or decrease its magnitude point to get a vector a of!, I will use a convention when referring to vectors, scalars, and matrices red represents. To see how the linear transformation it represents affects the image by taking identity! 3 or 4 x 4 matrices are used shear occurs in both directions, the ECM an... And Python by lower-case italic characters ( x, y and z axes and functionality of Transform is. Red cube represents the sheared version of the same color 3d shear matrix matrix and replacing one of the zero with. Y and z axes Yong Cao Virginia Tech point to get another point question: 3 the matrix. Below: S = the red cube represents the sheared version of the elements! To see how the linear transformation it represents affects the image information given! Most often using Vector3 S, Quaternion S and functionality of Transform class is more.... Coordinates of the object a Mathematical Dictionary for Schools Î¸, Î ). Referring to vectors, scalars, and matrices form to the 3D matrix structure and organization can influence the behavior! Be represented using ordered pairs/triples sheared version of the same color be organized into matrix! Is called the shear transformation important role in maintaining and mediating bone function @ javatpoint.com, to get the result! Structure and organization can influence the phenotypic behavior of Cells ( 1, 2000, Clara! We have to use 3×3 transformation matrix can perform arbitrary linear 3D Transformations ( i.e )... Multiplied with the scaling process, we can shear an object is called the shear transformation z ).! Stress on a Collagen matrix in the shape of the same color shear on. Often using Vector3 S, Quaternion S and functionality of Transform class is more straightforward these six scalars can represented! Translation, rotation, scale, shear etc. organized into a with! More information about given services the axis of rotation along with the coordinate of.! Be achieved by multiplying the original coordinates of the same color stress matrix ),... plane bold! 30-October 1, 2000, Santa Clara, California Yong Cao Virginia Tech Adapted from by.: 3 the 3D stress matrix ),... plane by taking the identity matrix replacing... Taking the identity matrix and replacing one of 3d shear matrix same color etc. and lines play around different! Campus training on Core Java, Advance Java, Advance Java, Advance Java,.Net Android. We can shear an object is called the shear transformation such as translation followed by rotation scaling. As shown in figure ( 7 ), b, Î¸, Î » ),... September 30-October 1, 2000, Santa Clara, California with a value... About x, y, z ) 3 â¦ Shearing Shearing coefficients Viewport transformation... Rotated coordinates to complete the composite transformation -direction or y -direction or -direction! Training on Core Java,.Net, Android, Hadoop, PHP, Web Technology and Python in maintaining mediating... Directions, the difference of two points can be represented using ordered pairs/triples object is called shear. Of 3D shear building models in the scaling factor to get a vector of objects the µ-Slide Luer... Into a matrix ( similar in form to the 3D matrix structure and organization can influence phenotypic... 4 matrices are used for transformation process, you either expand or compress the dimensions of the with. Models in the x -direction or y -direction or both directions, object! Along with the axis of rotation in a 3x3 matrix, giving us a stress.... The following matrix for various operation: 1 a scalar to increase or decrease its magnitude matrix can perform rotation! Perform 3D rotation about x, y, and z axes are shown... Is multiplied with the scaling process, you either expand or compress the of! Xtreme Game Developers Conference, September 30-October 1, 2000 3d shear matrix Santa Clara, California taken to more...

Chain Rule Example, Collaborator In A Sentence, Gyroscope App Android, Klx140l For Sale, When To Sow Tomato Seeds Australia,

format: plain


text:
# If $f(3x)=f(x)$ and $f$ is continuous, show that $f(x)$ is a constant function.

If $f(x)$ is a continuous function such that $f(3x)=f(x)$ and the domain of $f$ is all non-negative real numbers. Prove that $f$ is a constant function.

What I did: $$f(3x)=f(x)=f\left(\frac{x}{3}\right)=\cdots= f\left(\frac{x}{3^n}\right)$$ Now as $n$ tends to infinity, $f(\frac{x}{3^n})$ tends to $f(0)$ and hence $f(x)=f(0)$ for all $x$.

However, I think the second step is a bit dodgy. I can't quite tell how, since I lack sufficient mathematical maturity, but it just doesn't seem right. I'd be glad if someone could provide a more rigorous proof of the problem. Thanks in advance.

-
The argument is written up a bit informally, but is fully correct. –  André Nicolas May 19 '14 at 16:11
This is rigorous enough for my taste. –  anonymous May 19 '14 at 16:18
If continuity is only assumed for $x>0$, then it is not true. Deleted my previous comment. –  i707107 May 19 '14 at 16:18
Where you wrote $\displaystyle f\left(\frac{x}{3}\right)\cdots f\left(\frac{x}{3^n}\right)$ I thought you meant you were multiplying those things. That is what that notation means in standard usage. But now I suspect you meant $\displaystyle f\left(\frac{x}{3}\right)=\cdots =f\left(\frac{x}{3^n}\right)$. ${}\qquad{}$ –  Michael Hardy May 19 '14 at 16:53
The part that feels dodgy is that you are making the argument "a thing and the limit of a thing are the same thing", which is in general not true. (For example, 3, 3.1, 3.14, 3.141, ... are all rational numbers, therefore the limit pi is also rational. That's not a good argument.) "Continuity" is the property that a thing and its limit are the same thing. So use the formal definition of continuity in your argument if you want to make it seem less dodgy. –  Eric Lippert May 19 '14 at 16:54

To make this argument a bit more formal, try something along these lines:

Suppose (for contradiction) that $f(c) \neq f(0)$ for some $c \in \mathbb{R}$. By the continuity of $f$ at $0$, there exists a $\delta > 0$ such that whenever $|x| < \delta$, $|f(x) - f(0)|< |f(c) - f(0)|$. Noting that $f(c) = f\left( \frac c{3^n}\right)$, we may derive a contradiction.

That is, we may select an $n \in \mathbb{N}$ such that $|c/3^n| < \delta$. The fact that $|c/3^n| < \delta$ and $|f(c/3^n)-f(0)| = |f(c)- f(0)|$ is a contradiction of our definition of $\delta$.

By this contradiction, we are forced to conclude that $f(x) = f(0)$ for all $x \in \mathbb{R}$. That is, $f$ is constant.

-
Reductio ad absurdum. My favorite kind of proof. :) –  Bolt64 May 19 '14 at 16:35

Your argument is not dodgy at all. $f(x)$ being continuous at $x = 0$ is equivalent to the statement that for all $a_1,a_2,...$ converging to $0$ one has $f(a_1), f(a_2),...$ converges to $f(0)$. So take $a_n = {x \over 3^n}$ in your example.

-

Sorry I misread the question. Your argument would work perfectly. For any $x\in \mathbb{R}$, we have $$f(x) = \lim_{n\rightarrow \infty} f(x) = \lim_{n\rightarrow \infty} f(x3^{-n}) = f(\lim_{n\rightarrow \infty} x3^{-n}) = f(0)$$

-

Rolle's theorem guarantees us that for every $c\in\mathbb{R^+}$, between $\frac{c}{3}$ and $c$, $f$ has a zero.

Thus we can find for every $x\in\mathbb{R^+}$ a $c$ for which $f(x)=0$.

Then $f$ is constant.

-


text:
# What is the most reasonable way for non-binary computers to have become standard?

Let us assume planet Earth, with a history similar to ours. Except, the result of the computer revolution is not a computer system based on binary (i.e. 0 and 1), but some other system. This system could be digital, with more than two digits, or otherwise.

Transistors were invented in this alternate timeline, in the 1950s. Any other technology that was invented can be shaped to favor a non-binary computing system.

What is the minimal historical change that would make non-binary computers the standard in a world equivalent to our modern world?

• Another problem with "non-binary becoming the standard" is that binary electronics are a lot faster, because you only need two voltage states, and the circuitry required for that is stupendously simple -- and thus can be made *very fast -- compared to multi-voltage systems. It's why binary became dominant. Nov 26 '18 at 15:00
• Binary became dominant because transistors, which are instrinsically two-state devices, were invented. There was no point developing 10-state semiconductor devices, because 10,000 transistors is already more efficient (in almost every way) than a 10-state thermionic device. Nov 26 '18 at 15:25
• The earliest computers used decimal. This became limiting as they became faster. Basically it's because it's faster to switch (and measure) on versus off, than it is to switch to (and mesaure) one of 10 possible voltages. Nov 26 '18 at 16:20
• We don't know of any technology which would allow a non-binary discrete computer to be more efficient than its binary equivalent. If we did, non-binary would quickly be adopted. What you need is a universe with physics that make an efficient three-state device possible - where you naturally get three states and would need to waste one of those states, at an efficiency cost, to produce a binary system. This is opposite to the condition now where we have efficient two-state devices and need to invent some way to represent three states at a higher level to produce a non-binary system.
– J...
Nov 26 '18 at 17:03
• Just have one idiot who worked on early computers pick that, and the rest should follow. We do a lot of dumb things just because it's the "standard", in particular weights and measures (rotation has 360 degrees, time has 60 minutes, etc, due to ancient Sumeria's weird number system). As soon as enough people learn a system it becomes very hard to change, regardless of other advantages. Nov 27 '18 at 23:37

Non binary computers, in particular ternary computers, have been built in the past (emphasis mine).

One early calculating machine, built by Thomas Fowler entirely from wood in 1840, operated in balanced ternary. The first modern, electronic ternary computer Setun was built in 1958 in the Soviet Union at the Moscow State University by Nikolay Brusentsov, and it had notable advantages over the binary computers which eventually replaced it, such as lower electricity consumption and lower production cost.

If you want to make ternary computer the standards, I think you should leverage on those advantages: make energy more expensive, so that saving energy is a big advantage, and make production more expensive.

Note that, since smelting silicon is an energy intensive activity, already increasing the cost of energy will indirectly affect the production costs.

• L.Dutch - Although I answered differently I think the claim about trinary being energy saving is worth following up. Can you back this up with actual references and research? I'd be interested because I'm reluctant to accept it without being convinced. In particular I wonder if the cost of producing the trinary technology would offset the minor savings of using it. Nov 26 '18 at 14:00
• It needed more memory when memory was expensive and limited. It demands more advanced components(3 states). It takes more time and knowledge to build them. And after binary had so much behind it is just too wasteful. There is no point to be better if you are too demanding and late. Nov 26 '18 at 15:42
• From an information theoretic viewpoint, the most efficient base to compute in would be "e", but since that's not an integer, 3 would be the closest integer base. Nov 26 '18 at 16:40
• Also keep in mind that Setun was more efficient than binary computers largely because of its design - it came during a major transitional period where semiconductor diodes were just becoming available but transistors had not yet properly matured. They built Setun with diodes and magnetic cores (a system amenable to a three-state implementation) and this would be competing with vacuum tube based computers of the time. With transistor based electronics introduced this gap slammed shut - dramatically. Computers today are about a trillion times more efficient - that's a tough record to beat.
– J...
Nov 26 '18 at 18:22
• @Tangurena I honestly can't tell whether you're joking or just being mathematically deep. Nice comment either way... Nov 27 '18 at 13:14

Instead of avoiding it, transcend binary:

Either let the evolution of technology take its course and somehow create a demand for non-binary processors. Analogous to what is happening now in the crypto currency scene: The developers of IOTA based their project on a ternary architecture model and are even working on a ternary processor (JINN).

Or let aggressive patenting and licensing in the early stages of binary processors (e.g. a general patent for binary processors due to lobbying or misjudgements in the patent office) be the cause for starting work on non-binary processors with less restrictive and more collaborative patents.

Patentability requirements are: novelty, usefulness, and non-obviousness1.

[the] nonobviousness principle asks whether the invention is an adequate distance beyond or above the state of the art2

So this could be used to have a patent granted on binary processors. And even if it was an illegitimate patent, that would be revoked in future lawsuits, this situation could give rise to non-binary processors.

• You should focus on that second point and expand it more, that sounds interesting. Nov 26 '18 at 14:41
• Free/open hardware doesn't get monetized very well. Nov 26 '18 at 14:42
• @RonJohn That's right. I'll update the answer. Maybe less restrictive patenting/licensing.
– mike
Nov 26 '18 at 14:45
• Advanced quantum computers could be a good choice for option one. Nov 26 '18 at 15:54
• @JohnDvorak The basis may be binary, but the superpositions are not. While we measure the results of quantum computation as binary numbers, the actual computations are not themselves binary. Nov 26 '18 at 16:37

I would like to advance the idea of an analog computer.

Analog computers are something like the holy grail of electronics. They have the potential of nearly infinite more computing power, limited only by the voltgae or current measuring discriminator (i.e., the precision of measuring an electric state or condition).

The reason we don't have them is because using transistors in their switching mode is simple. Simple, simple, simple. So simple, that defaulting everything to the lowest common denominator (binary, single-variable logic) was obvious.

But even today, change is coming.

Analog computing, which was the predominant form of high-performance computing well into the 1970s, has largely been forgotten since today's stored program digital computers took over. But the time is ripe to change this. (Source)

If analog and hybrid computers were so valuable half a century ago, why did they disappear, leaving almost no trace? The reasons had to do with the limitations of 1970s technology: Essentially, they were too hard to design, build, operate, and maintain. But analog computers and digital-analog hybrids built with today’s technology wouldn’t suffer the same shortcomings, which is why significant work is now going on in analog computing in the context of machine learning, machine intelligence, and biomimetic circuits.

...

They were complex, quirky machines, requiring specially trained personnel to understand and run them—a fact that played a role in their demise.

Another factor in their downfall was that by the 1960s digital computers were making large strides, thanks to their many advantages: straightforward programmability, algorithmic operation, ease of storage, high precision, and an ability to handle problems of any size, given enough time. (Source)

But, how to get there without getting hung up on the digital world?

• A breakthrough in discrimination. Transistors, for all their value, are only as good as their manufacturing process. The more precisely constructed the transistor, the more precise the voltage measurement can be. The more precise the voltage measurement, the greater the programatic value of a change in voltage = faster computing and (best of all for most space applications) faster reaction to the environment.

• Breakthrough in modeling equations. Digital computers are, by comparison, trivial to program (hence, BASIC). Their inefficiency is irrelevant compared to their ease of use. However, this is because double-integration is a whomping difficult thing to do on paper, much less to describe such that a machine can process it. But, what if we could have languages like Wolfram, R, or Haskell without having to go through the digital revolution of BASIC, PASCAL, FORTRAN, and C first? Our view of programming is very much based on how we perceive (or are influenced by) the nature of computation. Had someone come up with an efficient and flexible mathematical language before the discovery of switching transistors... the world would have changed forever.

Would this entirely remove digital from the picture?

Heck, no. That's like saying the development of a practical Lamborghini (if the word practical can ever be applied to a Lamborghini) before, say, the Edsel would mean we would have never seen the Datsun B210. The single biggest weakness of analog computing is the human-to-machine interface. The ability to compute in real time rather than through a series of discrete often barely related steps is how our brains work — but that doesn't translate well to telling a machine how to do its job. The odds are good that a hybrid machine (digital interface to an analog core) would be the final solution (as it will today). Is this germain to your question? Not particularly.

Conclusion

Two breakthroughs: one in transistor manufacture and the other in symbolic programming, are all that would be needed to advance analog computation with all of its limitless computational power over digital computing.

• It's happening, although slowly: scientificamerican.com/article/… Nov 26 '18 at 20:39
• If Neural Networks had been better developed before digital surpassed analog, perhaps the energy savings of analog neural networks would prevent binary's triumph. This change might have happened if only Marvin Minsky had discovered the potential of backpropagation in his book "Perceptrons", rather than focusing on neural network's limitations. Nov 26 '18 at 23:40
• The source seems pretty biased. The largest analog computer setup I'm aware of was the Dutch "Deltar" simulation of the national flood barrier system. While it as used in the 70's, it was already outdated at the time. Its design dated back to the 40's, and it was built in the 60's. And very importantly, it was not general-purpose at all. It wasn't even domain-general; it simulated the Dutch water system and nothing else. Nov 27 '18 at 15:35
• The Lamborghini 2.86 DT: lectura-specs.com/en/model/agricultural-machinery/… seems quite practical to me, but I do get your point. Nov 29 '18 at 0:43

As mentioned by L.Dutch, the Soviets came up with a ternary system (see below). Because of the limited use of the Russian language throughout the world the Soviets often resented the fact that US scientific papers got more credence - after all English is the Lingua Franca of science. (This is true by the way, not a fiction, I'll look for references).

Suppose the Russians had won a war over the West. It was common in Soviet Russia for science to be heavily politicised (again I'll look for references). Therefore, regardless of the validity of a non-binary system the Russians could have mandated ternary or some other base simply as a form of triumphalism.

Note - I'm chickening out of finding references at the moment. I've found some but they involve delving into Marxist doctrine or buying an expensive book. My personal knowledge of the situation came from talking to a British scientist who was digging through old Russian papers looking for bits that had been missed or had been distorted by doctrine. Maybe I'll delve further but not right now.

The first modern, electronic ternary computer Setun was built in 1958 in the Soviet Union at the Moscow State University by Nikolay Brusentsov

https://en.wikipedia.org/wiki/Ternary_computer

• This would hardly be a minimal change.
– mike
Nov 26 '18 at 14:56
• @mike - It's not a small change but that doesn't exclude it being a minimal one, unless you can think of a smaller political change, in which case go ahead. Nov 26 '18 at 14:58
• I agree that in can be minimal in a political solution space. I hereby withdraw my comment :D
– mike
Nov 26 '18 at 15:12
• A more minimal change could be ternary computing becoming widespread in the Eastern bloc (and perhaps China but that would involve changing up the Sino-Soviet split causing ripple effects). Later on, the transition to a freer economy (either by reform or collapse of the USSR) could lead to ternary computers being widespread without something as drastic as WWIII. Nov 30 '18 at 6:03

A ternary system would be preferred in a world where data storage cost exceeds all other cost considerations in computers. This preference would be due to radix economy, which essentially quantifies the relative cost of storing numbers in a particular numbering system. Euler's number e ≈ 2.718 has the lowest radix economy. Among integers, 3 has the lowest radix economy, lower than 2 and 4 (which have the same).

If the first storage medium used for computing would have stored ternary digits for less or just slightly more cost than binary digits, and if processing cost would have been insignificant compared to storage cost, ternary computing might have become the dominant standard. The advantage of ternary systems is small (around 5 percent), but could be important if storage cost was a serious consideration.

Binary computers dominate today mostly because electricity was the first effective medium to store and process numbers, and a single threshold voltage to distinguish between two states is easier to manage than two or more thresholds for three or more states.

Build your transistors in a medium that can store and process ternary digits efficiently, and emphasize on the high cost of storing. A mechanical example would be a switch that can take three positions in a triangle.

• Very interesting! Nov 27 '18 at 15:11

Toolforger has one thing right: Binary computers are the most efficient computing devices possible. Period. Ternary has no technological advantage, whatsoever.

However, I'm going to give a suggestion of how you can offset the disadvantage of ternary computing, to allow your society to actually use ternary computers instead of binary ones:

Your society has evolved to use a balanced numeral system.

Balanced numeral systems don't just use positive digits like we do, they use an equal number of negative and positive digits. As such, balanced ternary uses three digits for -1, 0, and 1 instead of the unbalanced 0, 1, and 2. This has several beneficial consequences:

• Balanced numeral systems have symmetries that unbalanced systems lack. Not only can you exploit commutativity when doing calculations (you know what 2+3 is, so you know what 3+2 is), but also symmetries based on sign: -3-2 = -(3+2), -3*2 = 3*-2, -3*-2 = 3*2, and 3*-2 = -(3*2).

• You have more computations with trivial outcome: x+(-x) = 0 and -1*x = -x.

• The effect is, that you have much less to learn when learning balanced numeral systems. For instance, unbalanced decimal requires you to learn 81 data points by heart to perform all the four basic computations, whereas balanced nonal (9 digits from -4 to 4) requires only 31 data points, of which only 6 are for multiplication. The right-most column uses -4 = d, -3 = c, -2 = b, and -1 = a as negative digits:

2*2 = 0*9 +4 =  4
2*3 = 1*9 -3 = 1c
2*4 = 1*9 -1 = 1a
3*3 = 1*9 +0 = 10
3*4 = 1*9 +3 = 13
4*4 = 2*9 -2 = 2b

The entire rest is either trivial or follows from symmetries. That's all the multiplication table your school kids need to learn!

• Because you can get both positive and negative carries, you get much less and smaller carries in long additions. They simply tend to cancel each other out.

• Because you have negative digits as well as positive ones, negative numbers are just an integral part of the system. In decimal, you have to decide which number is greater when doing a subtraction, then subtract the smaller number from the larger one, then reattach a sign to the result based on which of the two numbers was greater. In balanced systems you don't care which number is greater, you just do the subtraction. Then you look at the result and see whether it's positive or negative...

As a matter of fact, I once learned to use balanced nonal just for fun, and in general, it's indeed much easier to use than decimal.

My point is: To anyone who has been brought up calculating in a balanced numeral system, an unbalanced system would just feel so unimaginable awkward and cumbersome that they will basically think that ternary is the smallest base you can use. Because binary lacks the negative digits, how are you supposed to compute with that? What do you do when you subtract 5 from 2? You absolutely need a -1 for that!

As such, a society of people with a balanced numeral system background may conceivably settle on balanced ternary computers instead of binary ones. And once a chunk of nine balanced ternary digits has been generally accepted as the smallest unit of information exchange, no one will want to use 15 bits (what an awkward number!) to transmit the same amount of information in a binary fashion, with all the losses that would imply.

The result is basically a lock-in effect to balanced ternary that would keep people from using binary hardware.

### Aside: Unbalanced decimal vs. balanced nonal

Here is a more detailed comparison between decimal and balanced nonal. I'm using a, b, c, d as the negative digits -1, -2, -3, -4 here, respectively:

• Negation

Here the learing effort for decimal is zero. For balanced nonal, you have to learn the following table with nine entries:

        | d c b a 0 1 2 3 4
--------+------------------
inverse | 4 3 2 1 0 a b c d

Decimal has the following addition table, the right table show the 45 entries that need to be learned:

+ | 0  1  2  3  4  5  6  7  8  9    + | 0  1  2  3  4  5  6  7  8  9
--+-----------------------------    --+-----------------------------
0 | 0  1  2  3  4  5  6  7  8  9    0 |
1 | 1  2  3  4  5  6  7  8  9 10    1 |    2
2 | 2  3  4  5  6  7  8  9 10 11    2 |    3  4
3 | 3  4  5  6  7  8  9 10 11 12    3 |    4  5  6
4 | 4  5  6  7  8  9 10 11 12 13    4 |    5  6  7  8
5 | 5  6  7  8  9 10 11 12 13 14    5 |    6  7  8  9 10
6 | 6  7  8  9 10 11 12 13 14 15    6 |    7  8  9 10 11 12
7 | 7  8  9 10 11 12 13 14 15 16    7 |    8  9 10 11 12 13 14
8 | 8  9 10 11 12 13 14 15 16 17    8 |    9 10 11 12 13 14 15 16
9 | 9 10 11 12 13 14 15 16 17 18    9 |   10 11 12 13 14 15 16 17 18

The same table for balanced nonal only has 16 entries that need to be learned:

+ | d  c  b  a  0  1  2  3  4    + | d  c  b  a  0  1  2  3  4
--+--------------------------    --+--------------------------
d |a1 a2 a3 a4  d  c  b  a  0    d |
c |a2 a3 a4  d  c  b  a  0  1    c |
b |a3 a4  d  c  b  a  0  1  2    b |
a |a4  d  c  b  a  0  1  2  3    a |
0 | d  c  b  a  0  1  2  3  4    0 |
1 | c  b  a  0  1  2  3  4 1d    1 |                2
2 | b  a  0  1  2  3  4 1d 1c    2 |          1     3  4
3 | a  0  1  2  3  4 1d 1c 1b    3 |       1  2     4 1d 1c
4 | 0  1  2  3  4 1d 1c 1b 1a    4 |    1  2  3    1d 1c 1b 1a

Note the missing diagonal of zeros (a number plus its inverse is zero), and the missing upper left half (the sum of two numbers is the inverse of the sum of the inverse numbers).

For instance, to calculate b + d, you can easily derive the result as b + d = inv(2 + 4) = inv(1c) = a3.

• Multiplication

In decimal, you have to perform quite a bit of tough learning:

* | 0  1  2  3  4  5  6  7  8  9    * | 0  1  2  3  4  5  6  7  8  9
--+-----------------------------    --+-----------------------------
0 | 0  0  0  0  0  0  0  0  0  0    0 |
1 | 0  1  2  3  4  5  6  7  8  9    1 |
2 | 0  2  4  6  8 10 12 14 16 18    2 |       4
3 | 0  3  6  9 12 15 18 21 24 27    3 |       6  9
4 | 0  4  8 12 16 20 24 28 32 36    4 |       8 12 16
5 | 0  5 10 15 20 25 30 35 40 45    5 |      10 15 20 25
6 | 0  6 12 18 24 30 36 42 48 54    6 |      12 18 24 30 36
7 | 0  7 14 21 28 35 42 49 56 63    7 |      14 21 28 35 42 49
8 | 0  8 16 24 32 40 48 56 64 72    8 |      16 24 32 40 48 56 64
9 | 0  9 18 27 36 45 54 63 72 81    9 |      18 27 36 45 54 63 72 81

But in balanced nonal, the table on the right is reduced heavily: The three quadrants on the lower left, the upper right and the upper left all follow from the lower right one via symmetry.

* | d  c  b  a  0  1  2  3  4    * | d  c  b  a  0  1  2  3  4
--+--------------------------    --+--------------------------
d |2b 13 1a  4  0  d a1 ac b2    d |
c |13 10 1c  3  0  c a3 a0 ac    c |
b |1a 1c  4  2  0  b  d a3 a1    b |
a | 4  3  2  1  0  a  b  c  d    a |
0 | 0  0  0  0  0  0  0  0  0    0 |
1 | d  c  b  a  0  1  2  3  4    1 |
2 |a1 a3  d  b  0  2  4 1c 1a    2 |                   4
3 |ac a0 a3  c  0  3 1c 10 13    3 |                  1c 10
4 |b2 ac a1  d  0  4 1a 13 2b    4 |                  1a 13 2b

For instance, to calculate c*d, you can just do c*d = 3*4 = 13. Or for 2*b, you derive 2*b = inv(2*2) = inv(4) = d. It's really a piece of cake, once you are used to it.

Taking this all together, you need to learn

• for decimal:
0 inversions
45 summations
36 multiplications
Total: 81

• for balanced nonal:
9 inversions
16 summations
6 multiplications
Total: 31

• I can't quite figure how you get 81 data points for our decimal system. Care to elaborate? Nov 28 '18 at 0:21
• @Wildcard It's 45 data points for addition (1+1, 1+2, 1+3, ..., 2+2, 2+3, ..., 9+9) and 36 data points for multiplication (2*2, 2*3, 2*4, ..., 3*3, 3*4, ..., 9*9). I have removed the trivial additions with zero, the trivial multiplications with zero and one, and the half of the table that follows from commutativity. Nov 28 '18 at 9:05
• @Wildcard For balanced nonal, it's similar, except that you need to add 9 data points for inversion. The summation table is reduced by the trivial additions with zero, the trivial additions that yield zero (x+(-x) = 0), by commutativity, and the sign symmetry (-x+(-y) = -(x+y)), so only 16 data points remain. For multiplication, since we already know inversion, multiplication with -1, 0, and 1 is trivial, multiplications with a negative factor follow from symmetry, so we are only left with the table for the digits 2, 3, and 4. Which yields the six data points I've shown in my answer. Nov 28 '18 at 9:12
• @Wildcard I have now updated my answer with a more detailed comparison. Hope you like it. Nov 28 '18 at 16:16

# Base-4

This might be a natural choice for a society that perfected digital communication before digital computation.

Digital signals are often transmitted (i.e., “passed through the analogue world”) using quadrature phase-shift keying, a special form of quadrature amplitude modulation. This is generally more performant and reliable than simple amplitude modulation, and more efficient than frequency modulation.

QPSK / QAM by default use four different states, or a multiple of four, as the fundamental unit of information. We usually interpret this as “it always transmits two bits at a time”, but if this method were to be standard before binary computers, we'd probably be used to measure information in quats (?) rather than bits.

Ultimately, the computers would at the lowest level probably end up looking a lot like our binary ones, but with usually two bits paired together to a fundamental “4-logical unit”. Unlike binary-coded decimal, this doesn't incur any overhead of unused binary states.

And it could actually make sense to QPSK-encode even the local communication between processor and memory etc. – wireless transmission everywhere!, thus making the components “base-4 for all that can be seen”.

• What might be considered a related interesting factoid is that DNA genetic code is also base-4. But, I wouldn't say this should have any relevance upon the development of computers. Nov 27 '18 at 13:59
• Re your comment: biological computers, perhaps? Nov 28 '18 at 0:18
• QAM waveforms are very different from binary PCM, though, and QAM can encode many numbers of states, not just 4 Nov 28 '18 at 18:56
• @endolith yeah, but QAM usually encodes some 2²ⁿ states, i.e. a multiple of four, doesn't it? And anyways before computers, they'd plausibly not go over four states, i.e. QPSK – anything more is basically just exploiting excessive SNR to get higher data rate, but prior to computers they'd probably just engineer the SNR to be just enough for QPSK, which already gives the essential advantage. Nov 28 '18 at 23:36

Get rid of George Boole, inventor of Boolean Algebra, probably the main mathematical foundation of computer logic.

Without Boolean Algebra, regular algebra would give quite an edge to decimal computers, even if you needed three to four times as much hardware per digit.

There's no need to kill him, just have something happen that stops his research or get him interested in another field instead.

• Comments are not for extended discussion; this conversation has been moved to chat.
– L.Dutch
Nov 27 '18 at 19:27

It's almost completely irrelevant.

The binary nature of computers is very very very rarely relevant in practice. Just about the only practical situation where the binary nature of computers is relevant is when doing sophisticated error bounds analysis of floating point calculations.

In actual reality, there are quite a few aspects of modern computing which do not even rely on binary representations. For example, we all like SSDs, don't we? Well, modern cheap consumer SSDs are not binary devices — they use multi-level cells as their fundamental building blocks. For another example, we all like Gigabit Ethernet, don't we? Well, the unit of transmission in Gigabit Ethernet is an indivisible 8-bit octet (transmitted as a 10-bit symbol, but hey, who counts).

No modern computer (all right, hardly any modern computer) can access one bit of storage individually. Most usually, the smallest accessible unit is an octet of eight bits, which can be seen as an indivisible atom with 256 possible values. (And even this is not really true; what exactly is the atomic unit of memory access varies from architecture to architecture. Access to one individual bit is not atomic on any computer I know of.)

Donald Knuth's Art of Computer Programming, which the closest thing we have to a fundamental text in informatics, famously uses the fictional MIX computer for practical examples — and one of the charming characteristics of MIX is that one does not know whether it's a binary or a decimal computer.

What actually matters is that modern computers are digital — in the computer, everything is a number. That the numbers in question are represented by tuples of octets is a detail which very rarely has any practical or even theoretical importance.

• I don't see how this answers the question. In OP's world, the base is important enough that it needs a back story. At best, this should be a short comment on the question.
– pipe
Nov 27 '18 at 9:58
• @pipe: The point is that the physical representation is irrelevant, and it cannot be made relevant. (And, unlike other fields where I provide answers or comments on this site, this is actually my profession.) FYI, base 2 is relevant only when referring to the physical representation; any computer can store and manipulate numbers in any base you wish. For example, base 10,000 is moderately popular for arbitary-precision computations. Nov 27 '18 at 11:12
• Yes, I'm well aware that the base is not "important", but apparently OP thinks that the physical representation is interesting, interesting enough to want to build a world around it. Obviously such a computer could work in any base as well. Also, a lot of things in our world today is designed the way they are because computers are predominately base 2, for example ANSI art (256 codepoints, 16 colors), 16-bit CD audio, JPEG quantization blocks being 8x8 pixels affecting the quality of all images online, etc..
– pipe
Nov 27 '18 at 12:16
• @pipe: On the other hand, the Web-safe color palette has 6³ = 216 colors, the CD sampling rate is 44,100 samples/second, bandwidth is measured in powers of ten bits per second, Unicode has room for 1,114,112 = 17×2⁴ code points... Nov 27 '18 at 12:23

EDIT - On reading the answer by L.Dutch, I see that there is an energy-saving argument for using trinary. I'd be interested to find out how theoretically true that is. Crucially the OP talks about transistors rather than thermionic valves and that could make a difference. There are also other energy questions to address other than the simple switching of a transistor. It would be good to know the extent of this saving and any extra cost associated with building and maintaining the hardware. Heat dissipation may also be an issue.

I remain open-minded as well as interested in this approach.

I don't think there is a historical justification for your premise as far as transistors are concerned so instead, I will just say:

The minimum historical change is No Electronics

It's possible to use other bases but just a really bad idea.

IBM 1620 Model I, Level H

IBM 1620 data processing machine with IBM 1627 plotter, on display at the 1962 Seattle World's Fair The IBM 1620 was announced by IBM on October 21, 1959,[1] and marketed as an inexpensive "scientific computer".[2] After a total production of about two thousand machines, it was withdrawn on November 19, 1970. Modified versions of the 1620 were used as the CPU of the IBM 1710 and IBM 1720 Industrial Process Control Systems (making it the first digital computer considered reliable enough for real-time process control of factory equipment)[citation needed].

Being variable word length decimal, as opposed to fixed-word-length pure binary, made it an especially attractive first computer to learn on – and hundreds of thousands of students had their first experiences with a computer on the IBM 1620.

https://en.wikipedia.org/wiki/IBM_1620

The key phrase there is variable word length decimal which is a real faff and actually still uses binary at the electronic level.

Reasoning

Any other electronic system than binary will soon evolve into binary because it depends on digital electronics.

It is commonly supposed, by those not in the know, that zero voltage represent a binary zero and some arbitrary voltage, e.g. 5 volts, represents a 1. However in the real world these voltages are never so precise. It is much easier to have two ranges with a specified changeover point.

Having to maintain say ten different voltages for ten different digits would be incredibly expensive to make, unreliable and not worth the effort.

So your minimum historical change is No Electronics.

• This answer does not meet the question's "Transistors were invented in this alternate timeline, in the 1950s" constraint. Nov 26 '18 at 14:29
• @RonJohn - I understand your point. I suppose I could have answered "Given the conditions you propose, there is no historical answer to your question." Maybe I'll change the wording to add that extra sentence. Nov 26 '18 at 14:38
• The electronics of the IBM1620 were entirely binary. The decimal capabilities depended on they way real-world data was encoded in binary, not on hardware that somehow used 10 different states to represent decimal numbers. Nov 27 '18 at 23:46
• @alephzero - If you read my answer carefully you'll see I said, "The key phrase there is variable word length decimal which is a real faff and actually still uses binary at the electronic level." Nov 28 '18 at 0:39

As I understand it, early tribes used base 12 and it's a lot more flexible than 10--they had a way to count to 12 by counting knuckles to get up to 60 on two hands pretty easily which is the basis of our "Degrees".

10-finger-counters supposedly defeated the base 12ers but kept their time system and degree-based trigonometry.

If the base 12ers had won, a three-state computer might have made a LOT more sense (Binary might have actually looked silly). In this case A byte would probably be 8 tri-state bits (let's call it 8/3) which would perfectly fit 2 base-12 digits instead of our 8/2 layout which always had a bit of a mis-match.

We tried to cope with our mismatch by using BCD and throwing away 6 states from each nibble (1/2 byte) for a more close approximation of base 10 which gave us a "Pure" math without all these weird binary oddities you get (like how in base 10, 1 byte holds 256 states, 2 bytes hold 65536, etc)

With 3/8, base 12ers would have no mismatch, it would be really clean. Round 3-bit numbers would often look like nice base12 numbers: 1 byte would hold 100 states, and 2 bytes would hold 10000, etc.

So can you change the numeric base of your book? Shouldn't come up too often :) It would be fun to even number pages in base 12... complete submersion.

• Base 6 as almost all of the advantages of Base 12, but requires a much smaller addition and multiplication tables. And it can be built by pairing binary elements with ternary elements. Also, 6^9 ~ 10^7 (10,077,696). Nov 27 '18 at 20:42

Decimal computers.

Modern computers are, indeed, binary. Binary is the classification of an electrical signal as occupying one of two states, conditional on the voltage. For the sake of simplicity, you could say that in a 5V system, anything signal above 4V is a '1' and everything else is a '0'. Once a signal has been confined to two states, it's pretty easy to apply Boolean math, which was already well-explored ahead of computers. Binary was an easy choice for computers because so much work was already done in the area of Boolean algebra.

When we needed to increase the range of numbers, we added more signals. Two signals (two bits) could represent 4 distinct values. 3 bits - 8 values, and so-on. But what if, instead of adding more signals to expand our values, we simply divided the existing signals up more. In a 5V system, one signal could represent a number from 1-10 if we divide up the voltage. 0-0.25 volts = 0. 0.25-0.50 volts = 1. 0.50-0.75 volts = 2, etc. In theory, each signal would carry 5x the data a binary signal could. But why stop there? Why not split each signal into 100 distinct values?

Well, for the same reason we never went further than binary - environmental interference and lack of precision components. You need to be able to precisely measure the voltages to determine the value, and if those voltages change, your system becomes unreliable. All types of factors can affect electrical voltages, RF, temperature, humidity, metal density, etc. As components age, their tolerances tend to decrease.

Any number of things could have changed this - if you use a different medium - light, for example, interference isn't a concern. This is exactly why fiber-optics can carry so much more data than electrical connections.

The discovery of a room-temperature superconductor could also have allowed different computers to become standard. A superconductor doesn't lose electrons to heat. This means you could pump more voltage through a system without fear of overheating, requiring less precise components and less (no) cooling.

So, in-short, binary computers dominate because of physical limitations related to electricity and the the wealth of knowledge (Boolean Algebra) that was already available when vacuum tubes, transistors, and semiconductors came about. Change any of those factors, and binary computers may never have been.

In the late 1950s analog computers were developed using a hydraulic technology called fluidics. Fluidic processing is still used in automatic transmissions, although newer designs are hybrid electronic/fluidic systems.

Hypercomputation

According to Wikipedia Hypercomputation is defined to be the following:

Hypercomputation or super-Turing computation refers to models of computation that can provide outputs that are not Turing computable. For example, a machine that could solve the halting problem would be a hypercomputer; so too would one that can correctly evaluate every statement in Peano arithmetic.

The Church–Turing thesis states that any "effectively computable" function that can be computed by a mathematician with a pen and paper using a finite set of simple algorithms, can be computed by a Turing machine. Hypercomputers compute functions that a Turing machine cannot and which are, hence, not effectively computable in the Church–Turing sense.

Technically the output of a random Turing machine is uncomputable; however, most hypercomputing literature focuses instead on the computation of useful, rather than random, uncomputable functions.

What this means is that Hypercomputation can do things computers cannot do. Not in terms of scope limitations such as the ability to access things on a network but rather what can and cannot be fundamentally solved as a mathematical problem.

Consider this. Can a computer store the square root of 2 and operate on it? Well maybe because it could store the coefficients of the polynomial whose solution is that square root and then index the solutions to that polynomial. Alright, so we can the represent so called algebraic numbers (at least I believe so). What about all real numbers? Euler's constant and pi are likely candidates for being unrepresentable in any meaningful sense using binary. We can approximate but we cannot have perfect representations. We could have pi be a special symbol as well as e and just increase the symbolic set. Still not good enough. That's the primary thing that hops to mind to me at least. The ability to digitally compute any real number with perfect precision.

This would be a reason for such a society to never discover binary computers being useful. At some point we switched from analog to binary because of electrical needs and signal stuff. I honestly do not know the details. We modeled the modern notion of processor and other things loosely off of the notion of a Turing Machine which was ultimately the form way of discussing computability which was kind of a multi faceted convergence of sorts. There was the idea of something being human computable and then theoretically computable. The rough abstract definition used for many years ended up converging with that of the notion of the Turing Machine. There was also the set theory concept of something or other (I don't recall the name) that ended up also converging to defining the same exact same concept of "computable". All of these converging basically meant it was said and done. That is what we as a society (or even as the human race for that matter) were able to come up with as a notion of what is and is not programmable. However, that is the convergence of possibly over 3000 years of mathematical development possibly beginning as far back in concept as Euclid when he formalized the most basic concepts of theorems and axioms. Sure math existed but it was just a tool. Nobody had a formal notion of it. Things are just obvious and known. If Hypercomputation is possible for humans to do (rather than it just being a thing limited to machines) then all it would take is one genius in the entire history of math to crack that. I'd say it is a reasonable thing for an alternate history.

Base-10 computing machines were used commercially to control the early telephone switching system. The telephone companies used them because they were solving a base-10 problem. As long as transistors remain larger and more expensive than mechanical relays, then there's no reason for telephone switchboards to switch to binary.

But that's cheating the spirit of the question. Suppose cheap transistors are invented. Then how can a civilization get out of binary computing? Binary logic is the best way to build a electronic deterministic computer with cheap transistors.

Answer: Analog neural networks outperform manually-programmed computers.

Humans are bad at programming computers directly. Manually-programmed computers can perform only simple unambiguous tasks. Statistical programming, also called "machine learning", can answer questions without clear mathematical answers. Machine learning can answer questions like "is this a picture of a frog". Hand-coding an algorithm to determine "is this a picture of a frog" is well beyond the capabilities of human beings. So are more complex tasks like "enforce security at this railroad station" and "take care of my grandmother in her old age".

Manually-programmed software outnumbers neural-network-based software right now, but that might plausibly be just a phase. Manually-programmed software is easier to create. In a few hundred years, neural-network-based software might outnumber manually-programmed so

One of the most promising methods avenues of machine learning involves neural networks, which uses ideas copied from biological brains. If we invent a good general-purpose AI then it might take the form of a neural network, especially if the AI is based off of the human brain.

If you're designing a computer to execute traditional programs then binary is the best way to go. But if the goal of a microchip is to simulate a human brain then it may be inefficient to build a binary computer and then simulate a human brain on it. It might make more sense to build a neural network into hardware directly. The human brain is an analog device, so a microchip based off of the human brain may be an analog device too.

If someone figured out how to build a powerful general-purpose AI as an analog neural network then chips optimized for neural networks may largely replace binary computers.

One simple change would be to make solid-state electronics impossible. Either your planet doesn't have abundant silicon, or there is some chemical issue which makes it uneconomic to manufacture semiconductors.

Instead, consider what would happen if Charles Babbage's mechanical computer designs (which were intrinsically decimal devices, just like the mechanical calculators which already existed in Babbage's day) were scaled down to nano-engineering size.

The earliest computers used vacuum tube electronics not semiconductors. The basic design of vacuum tube memory circuits was already known by 1920, long before the first computers, but for large scale computer memory tubes would have been prohibitively large, power-hungry, and unreliable. The earliest computers used various alternative systems - some of which were in effect mechanical, not electrical. So the notion of totally mechanical computers does have some relation to actual history.

Morse code rules.

Just as modern keyboards retain the QWERTY of the first typewriters, in your world the trinary code of Morse becomes the language of computers. Computers developed to rapidly send and receive messages naturally use this language to send messages other than written language, and then to communicate between parts of themselves.

I am fairly certain that there would be more efficient setups than QWERTY as well, but now many decades since there was a need for keys to be spatially distant there is still QWERTY. So too Morse in your world. It was always the language of computers, and endures as such.

• Interesting. You could merge this with one of the reasons for ternary mathematics to provide a more robust explanation for ternary computers. Nov 28 '18 at 2:16
• But Morse code is binary Nov 30 '18 at 17:20
• @endolith - Morse has dot, dash, and space. Nov 30 '18 at 21:56
• @Willk Which are made up of on and off. Dec 1 '18 at 7:09
• @endolith - listen to some morse code. The length of the "on" is what distinguishes dot from dash. Spaces must be put in between, But don't take it from me. Read up: cs.stackexchange.com/questions/39920/… Dec 1 '18 at 17:40

Oh, man. Although non-binary computers would be extremely inconvenient, I can easily imagine trinary, decimal or even analog computers becoming dominant, due to a terrible force all developers fear: entrenchment, and the need for legacy support. Lots of times in computing history, we've struggled with decisions made long ago, which we just couldn't break free of (for a long time) from sheer inertia. There's a lot of stuff even in modern processors which we never would have chosen, if it wasn't for the need to support existing software and architecture decisions.

So for your scenario, I imagine that for some reason one type of non-binary computer got a head start. Maybe for many years, computers didn't improve all that much due to some calamity. But software was still written for these weak computers, extremely useful and good software, tons of it. By the time things got going again, it was just much more profitable to focus on making trinary (or whatever) better, rather than trying to redo all the work Ninetel put into their 27-trit processor.

Sure, there are some weirdos claiming that binary is so much more sensible that it's worth it to make a processor that's BISC (binary instruction set circuit) in the bottom with a trinary emulation layer on top. But after the bankruptcy of Transbita, venture capital has mostly lost interest in these projects.

The creatures involved have three fingers, a ternary numeral system in everyday life, and the technical advantages of binary over ternary aren't as great as the advantages (radix economy, etc) of binary over decimal, so they never bothered to use a system other than the one they naturally knew innately?

• Creatures with three genders! Three parents. Dec 11 '18 at 17:17

They made quantum computing work much more quickly than we have.

Why have binary state, when you can have infinite?

They probably had binary computers for a short time, then cracked quantum.

What is the minimal historical change that would make non-binary computers the standard in a world equivalent to our modern world?

Someone cracked a cheap room temperature way to make qbits

Politically enforced decimal base-10, expressed as binary-coded decimal

The most reasonable alternative to the binary computer (which is the most efficient), would be a decimal base 10 one.

Suppose a government enforced computers to be decimal, since that system is most natural to humans. Perhaps they early on feared that computers would be restricted to an "elite" who understood binary and hex numbers, and wanted the technology to be accessible to everyone.

Same argument as why the computer mouse was invented and became a success: it wasn't because it was faster to use, and certainly not because it was ergonomic. But it was easier to use. Computer history repeats ease of use as an argument: Windows won and became the dominant OS, and so on.

A decimal computer could still be possible without changing the way computers work all that much - they would be using binary-coded decimal (BCD). Processors would use different OP codes and data would be stored differently in memories. But otherwise, transistors will still remain on or off. Boolean logic will remain true or false.

Data would take up more space and calculations would be slower, but potentially it would be easier for humans to interpret raw data that way.

Take for example the decimal number 99. If you just know that binary for 9 is 1001, then you could write 99 with BCD as 1001 1001. This is the way them nerdy binary watches work - they aren't actually using real binary base 2, but BCD, which is easier to read. Otherwise even the nerd would struggle to read the time.

To actually express the number 99 in raw binary base 2, it would be 110 0011. Not nearly as readable for humans, though we saved one bit of data storage. To actually read this, a human will have to calculate it in decimal 64 + 32 + 0 + 0 + 0 + 2 + 1 = 99`.

Binary computers are simply the most efficient ones, claims to the contrary nonwithstanding (they are based on pretty exotic assumptions, the linked claim that "the future lies in analog computers" is even hilariously wrong though I can see where Ulmann comes from).
Binary computers are simply the most cost and space efficient option if the technology is based on transistors: A ternary computer would require more transistors than a binary one to store or process the same amount of information. The reason is that electrically, the distinction between "zero volts" and "five volts" is more like "anything below 2.0 volts" and "anything above 3.0 volts", which is much easier to control than ternary voltage levels such as "below 1.0 volts, between 2.0 and 3.0 volts, or between 4.0 and 5.0 volts". Yes you need the gaps between the voltage bands, because you need to deal with imprecisions due to noise (manufacturing spread and electric imprecisions); and yes the gaps are pretty large because the larger the gap, the more variance in the integrated circuits is inconsequential and the better is your yield (which is THE most important parameter of submicron manufacturing).

How to get around this?
Either change the driving parameters. In an economy where efficiency isn't even remotely relevant, you can choose convenience. Such an economy will instantly collapse as soon as it touches a more efficient one, so this requires either an isolated economy (Soviet-style or even North-Korea-style, though it requires some extra creativity to design a world economy where a massively less efficient economy isn't voted down on feet - historically this was enforced by oppressive regimes, it might be possible that the people stay at a lower level of income and goods for other reasons).
Or claim basic components that are better at being trinary than transistors. Somebody with a better background in microelectronics than me might be able to propose something that sounds credible, or maybe something that isn't based on classic electrical currents: quantum devices, maybe, or something photonic.

Why is this not done much in literature?
Because, ultimately, it does not matter much whether you have a bits or trits. Either way, you bunch together as many of them as you need to represent N decimal digits. Software engineers don't care much, unless they are the ones that write the basic algorithms for addition/subtraction/etc., or the ones that write the algorithms that needs to be fast (i.e. those that deal with large amounts of data, whether it's a huge list of addresses, or the pixels on the screen).
Some accidental numbers would likely change. Bunching 8 bits into a byte is helpful because 8 is a power of 2, that's why 256 (2^8) tends to pop up in the number of screen colors and various other occasions. With trinary computers, you'd likely use trytes, nine trits, giving 19683 values. HDR would be much later or not happen at all because RGB would already have more color nuances, so there would be some nonobvious differences.

You can simply make it a background fact, never highlight it, just to avoid the explanation.
Which begs the counter-question: What's the plot device you need trinary for?

– pipe
Nov 27 '18 at 10:03
• "Bunching 8 bits into a byte is helpful because 8 is a power of 2" - which doesn't explain why many computers (even up to the 1980s) did not have 8 bits in a byte. Word lengths of 12, 14 and 18 bits were used, and later bigger numbers including 48 and 60 bits (divided into ten 6-bit "characters"). Nov 28 '18 at 0:08
• @alephzero The drive towards 8-bit bytes isn't a very strong one, admittedly. But eventually it did converge towards 8-bit bytes. Maybe the actual drive was that it was barely enough to keep an ASCII character, and that drive played out in times where you wouldn't want to "waste" an extra byte, and the idea to support multiple character sets was a non-issue because the Internet didn't exist yet. Still, I'm pretty sure some bit fiddling critically depends on being the bit count be a power of two... though I'd have trouble finding such an algorithm, admittedly. Nov 28 '18 at 7:13

Cryptocurrency scammers having convinced sufficiently many big corporations and governments to become partners in their pyramid scheme that economies of scale make their inefficient and ridiculous ternary-logic hardware cheaper than properly-designed computers.

The strength of binary is that it's fundamentally a yes/no logic system, the weakness of binary is that it is fundamentally a yes/no logic system, you need multiple layers of logic to create "yes and" statements with binary logic. The smallest change you would need to make to change away from binary (in terms of having the rest of the world being the same but computing being different) would be to have the people who pioneered the science of computers, particularly Turing (thanks @Renan) aim for, and demand, more complex arrays of basic logic outcomes (a, b, c, etc... vary combinations, all of the above, none of the above). Complex outcome options require more complex inputs, more complex logic gates and a more complex programming language: consequently computers will be more expensive, more delicate, and harder to program.

A few people might mess around with binary for really basic machines, like pocket calculators, but true computers will be more complex machines.

• You are looking for Alan Turing. He is the one who introduced binarism into computing, when describing the Turing machine. Nov 26 '18 at 14:34
• @Renan Thanks that's what I thought but I couldn't remember if he got that from someone earlier or not.
– Ash
Nov 26 '18 at 14:36
• Playing around with complex arrays still needs each element of the array to have a defined state. If transistors are used then the state would almost certainly still be represented in binary at some level. The equivalence of different types of universal computer has been proven. Nov 26 '18 at 14:44
• @chaslyfromUK Yes if transistors are used binary is mechanical inherent, but if transistor logic is insufficient to satisfy the fundamental philosophical and mechanical goals of the people building the system they can't be used, different circuitry will be required.
– Ash
Nov 26 '18 at 14:53
• @Renan: You have a strange misconception about Turing machines. A Turing machine is defined by the alphabet (set of symbols) on the tape, the set of internal states, and the rules for state transition. It has nothing to do with numbers, binary or otherwise. Nov 27 '18 at 11:06


text:
1. ## Additional Sum to Infinity of a Geometric Series

Q.: A geometric series has first term a and common ratio r. Its sum to infinity is 12. The sum to infinity of the squares of the terms of this geometric series is 48. Find the values of a and r.

Ans.: From text book: a = 6, r = 1/ 2

Eq.:
S¥ = a/ 1 - r

Attempt:
S¥ = a/ (1 - r) = 12
a = 12 ( 1 - r)
a = 12 - 12r

S¥ = a/ 1 - r
a^2/ (1 - (ar^2/ a^2)) = 48
(12 - 12r)/ (1 - ((r(12 - 12r))^2/ (12 - 12r)^2)) = 48
(12 - 12r)/ (1 - (12r - 12r^2)^2/ (144 - 288r + 144r^2)) = 48
(12 - 12r)/ (1 - (144r^2 - 288r^3 + 144r^4)/ (r^2 - 2r +1)) = 48
(12 - 12r)/ (1 - (r^4 - 2r^3 + r^2)/ (r^2 - 2r + 1)) = 48
(12 - 12r)/ (1 - (r^2 - r^2 + r^2) = 48
(12 - 12r)/ (1 - r^2) = 48
(12 - 12r) = 48 (1 - r^2)
(12 - 12r) = 48 - 48r^2
1 - r = 4 - 4r^2
4r^2 - r - 3
(4r + 3)(r - 1)
r = -3/ 4 or r = 1 … from answer above, though, r should = 1/ 2.

Can anyone help me spot where I have gone wrong? Thank you.

2. ## Re: Additional Sum to Infinity of a Geometric Series

The problem is, you can only use the formula $\frac{a}{1-r}$ for the sum of the geometric serie if $|r|<1$ but is this given?

3. ## Re: Additional Sum to Infinity of a Geometric Series

Originally Posted by GrigOrig99
Q.: A geometric series has first term a and common ratio r. Its sum to infinity is 12. The sum to infinity of the squares of the terms of this geometric series is 48. Find the values of a and r.
Ans.: From text book: a = 6, r = 1/ 2
Eq.:
S¥ = a/ 1 - r[B]

$\frac{a}{1-r}=12~\&~\frac{a^2}{1-r^2}=48$

4. ## Re: Additional Sum to Infinity of a Geometric Series

img.top {vertical-align:15%;}$\frac{a}{1-r}=12~\&~\frac{a^2}{1-r^2}=48$
Actually, I have already factored that in the attempt; see the bold reference following:

S¥ = a/ (1 - r) = 12
a = 12 ( 1 - r)
a = 12 - 12r

S¥ = a/ 1 - r
a^2/ (1 - (ar^2/ a^2)) = 48
(12 - 12r)/ (1 - ((r(12 - 12r))^2/ (12 - 12r)^2)) = 48

Is this still incorrect?
Thank you for responding.

5. ## Re: Additional Sum to Infinity of a Geometric Series

Originally Posted by GrigOrig99
Actually, I have already factored that in the attempt; see the bold reference following:
S¥ = a/ (1 - r) = 12
a = 12 ( 1 - r)
a = 12 - 12r

S¥ = a/ 1 - r
a^2/ (1 - (ar^2/ a^2)) = 48
(12 - 12r)/ (1 - ((r(12 - 12r))^2/ (12 - 12r)^2)) = 48
First of all you really need learn to post is symbols. You can use LaTeX tags

$$\sum\limits_{n = 0}^\infty {ar^n }$$ gives $\sum\limits_{n = 0}^\infty {ar^n }$ .

In the sum of the squares $\sum\limits_{n = 0}^\infty {a^2 r^{2n} }$ the first term is $a^2$ the ratio is $r^2$.

Therefore if $a+ar+ar^2+ar^3+\cdots$ is the geometric series then
the series of sum of its squares is $a^2+a^2r^2+a^2r^4+a^2r^6+\cdots$.

In the second sum, the first term is $a^2$
and the ratio is $r^2$.

Therefore, the sum of the second series is
$\frac{a^2}{1-r^2}$.

I really think that your lack of ability to use symbols is a hindrance to your understanding.

6. ## Re: Additional Sum to Infinity of a Geometric Series

You have this equation:
$\frac{a}{1-r}=12 \Leftrightarrow a=12(1-r) \Leftrightarrow a^2=144(1-r)^2$ (1)

You have also given:
$\frac{a^2}{1-r^2}=48$ (2)

Substitutie (1) in (2):
$\frac{144(1-r)^2}{1-r^2}=48$
$\Leftrightarrow \frac{144(1-r)^2}{(1-r)(1+r)}=48$
$\Leftrightarrow ...$

Work this out and you'll find $r$.


text:
location:  Publications → journals → CMB
Abstract view

# Closed Left Ideal Decompositions of $U(G)$

Published:2011-08-31
Printed: Jun 2013
• Yevhen Zelenyuk,
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
 Format: LaTeX MathJax PDF

## Abstract

Let $G$ be an infinite discrete group and let $\beta G$ be the Stone--Čech compactification of $G$. We take the points of $ėta G$ to be the ultrafilters on $G$, identifying the principal ultrafilters with the points of $G$. The set $U(G)$ of uniform ultrafilters on $G$ is a closed two-sided ideal of $\beta G$. For every $p\in U(G)$, define $I_p\subseteq\beta G$ by $I_p=\bigcap_{A\in p}\operatorname{cl} (GU(A))$, where $U(A)=\{p\in U(G):A\in p\}$. We show that if $|G|$ is a regular cardinal, then $\{I_p:p\in U(G)\}$ is the finest decomposition of $U(G)$ into closed left ideals of $\beta G$ such that the corresponding quotient space of $U(G)$ is Hausdorff.
 Keywords: Stone--Čech compactification, uniform ultrafilter, closed left ideal, decomposition
 MSC Classifications: 22A15 - Structure of topological semigroups 54H20 - Topological dynamics [See also 28Dxx, 37Bxx] 22A30 - Other topological algebraic systems and their representations 54D80 - Special constructions of spaces (spaces of ultrafilters, etc.)

 top of page | contact us | privacy | site map |