text:
## FANDOM

64.590 Pages

Ισοτροπική Ομάς

isotropy Group, stabilizer group

- Ένα είδος Ομάδων.

## ΕτυμολογίαEdit

Η ονομασία "ισοτροπική" σχετίζεται ετυμολογικά με την λέξη "ισοτροπία".

## ΕισαγωγήEdit

In mathematics, an action of a group is a way of interpreting the elements of the group as "acting" on some space in a way that preserves the structure of that space.

Common examples of spaces that groups act on are:

Actions of groups on vector spaces are called representations of the group.

Some groups can be interpreted as acting on spaces in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another element of the set.

More generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions.

For other groups, an interpretation of the group in terms of an action may have to be specified, either because the group does not act canonically on any space or because the canonical action is not the action of interest.

For example, we can specify an action of the two-element cyclic group $\mathrm{C}_2 = \{0,1\}$ on the finite set $\{a,b,c\}$ by specifying that

• 0 (the identity element) sends $a\mapsto a,b\mapsto b,c\mapsto c$, and that
• 1 sends $a\mapsto b,b\mapsto a,c\mapsto c$.

This action is not canonical.

A common way of specifying non-canonical actions is to describe a homomorphism $\varphi$ from a group G to the group of symmetries of a set X.

The action of an element $g\in G$ on a point $x\in X$ is assumed to be identical to the action of its image $\varphi(g) \in \text{Sym}(X)$ on the point $x$.

The homomorphism $\varphi$ is also frequently called the "action" of G, since specifying $\varphi$ is equivalent to specifying an action. Thus, if G is a group and X is a set, then an action of G on X may be formally defined as a group homomorphism $\varphi$ from G to the symmetric group of X. The action assigns a permutation of X to each element of the group in such a way that:

If X has additional structure, then $\varphi$ is only called an action if for each $g\in G$, the permutation $\varphi(g)$ preserves the structure of X.

The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them.

In particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.

## Definition Edit

If G is a group and X is a set, then a (left) group action φ of G on X is a function

$\varphi\colon G \times X \to X\colon (g,x)\mapsto \varphi(g,x)$

that satisfies the following two axioms (where we denote φ(g, x) as g.x):[1]

Identity
e.x = x for all x in X. (Here, e denotes the identity element of the group G.)
Compatibility
(gh).x = g.(h.x) for all g, h in G and all x in X. (Here, gh denotes the result of applying the group operation of G to the elements g and h.)

The group G is said to act on X (on the left). The set X is called a (left) G-set.

From these two axioms, it follows that for every g in G, the function which maps x in X to g.x is a bijective map from X to X (its inverse being the function which maps x to g−1.x). Therefore, one may alternatively define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to X.[2]

In complete analogy, one can define a right group action of G on X as an operation X × GX mapping (x, g) to x.g and satisfying the two axioms:

Identity
x.e = x for all x in X.
Compatibility
x.(gh) = (x.g).h for all g, h in G and all x in X;

The difference between left and right actions is in the order in which a product like gh acts on x. For a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. Because of the formula (gh)−1 = h−1g−1 , one can construct a left action from a right action by composing with the inverse operation of the group. Also, a right action of a group G on X is the same thing as a left action of its opposite group Gop on X. It is thus sufficient to only consider left actions without any loss of generality.

## Types of actions Edit

The action of G on X is called

• Transitive if X is non-empty and if for each pair x, y in X there exists a g in G such that g.x = y. For example, the action of the symmetric group of X is transitive, the action of the general linear group or the special linear group of a vector space V on V ∖ {0} is transitive, but the action of the orthogonal group of a Euclidean space E is not transitive on E ∖ {0} (it is transitive on the unit sphere of E, though).
• Faithful (or effective) if for every two distinct g, h in G there exists an x in X such that g.xh.x; or equivalently, if for each ge in G there exists an x in X such that g.xx. In other words, in a faithful group action, different elements of G induce different permutations of X. In algebraic terms, a group G acts faithfully on X if and only if the corresponding homomorphism to the symmetric group, G → Sym(X), has a trivial kernel. Thus, for a faithful action, G embeds into a permutation group on X; specifically, G is isomorphic to its image in Sym(X). If G does not act faithfully on X, one can easily modify the group to obtain a faithful action. If we define N = {g in G : g.x = x for all x in X}, then N is a normal subgroup of G; indeed, it is the kernel of the homomorphism G → Sym(X). The factor group G/N acts faithfully on X by setting (gN).x = g.x. The original action of G on X is faithful if and only if N = {e}.
• Free (or semiregular or fixed point free) if, given g, h in G, the existence of an x in X with g.x = h.x implies g = h. Equivalently: if g is a group element and there exists an x in X with g.x = x (that is, if g has at least one fixed point), then g is the identity. Note that a free action on a non-empty set is faithful.
• Regular (or simply transitive or sharply transitive) if it is both transitive and free; this is equivalent to saying that for every two x, y in X there exists precisely one g in G such that g.x = y. In this case, X is called a principal homogeneous space for G or a G-torsor. The action of any group G on itself by left multiplication is regular, and thus faithful as well. Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G). This result is known as Cayley's theorem.
• n-transitive if X has at least n elements and for all pairwise distinct x1, ..., xn and pairwise distinct y1, ..., yn there is a g in G such that g.xk = yk for kn. A 2-transitive action is also called doubly transitive, a 3-transitive action is also called triply transitive, and so on. Such actions define interesting classes of subgroups in the symmetric groups: 2-transitive groups and more generally multiply transitive groups. The action of the symmetric group on a set with n elements is always n-transitive; the action of the alternating group is n-2-transitive.
• Sharply n-transitive if there is exactly one such g.
• Primitive if it is transitive and preserves no non-trivial partition of X. See primitive permutation group for details.
• Locally free if G is a topological group, and there is a neighbourhood U of e in G such that the restriction of the action to U is free; that is, if g.x = x for some x and some g in U then g = e.

Furthermore, if $G$ acts on a topological space $X$, then the action is:

• Wandering if every point $x \in X$ has a neighbourhood $U$ such that $\{g \in G : g \cdot U \cap U \neq \emptyset\}$ is finite.[3] For example, the action of $\mathbb Z^n$ on $\mathbb R^n$ by translations is wandering. The action of the modular group on the Poincaré half-plane is also wandering.
• Properly discontinuous if $X$ is a locally compact space and for every compact subset $K \subset X$ the set $\{g \in G: gK \cap K \neq \emptyset \}$ is finite. The wandering actions given above are also properly discontinuous. On the other hand, the action of $\mathbb Z$ on $\mathbb R^2 \setminus \{0\}$ by the linear map $(x, y) \mapsto (2x, 1/2y)$ is wandering and free but not properly discontinuous.Πρότυπο:Sfn
• Proper if $G$ is a topological group and the map from $G \times X \rightarrow X \times X : (g,x) \mapsto (g \cdot x,x)$ is proper.[4] If G is discrete then properness is equivalent to proper discontinuity for G-actions.
• Said to have discrete orbits if the orbit of each $x \in X$ under the action of $G$ is discrete in $X$.[3]

If X is a non-zero module over a ring R and the action of G is R-linear then it is said to be

• Irreducible if there is no nonzero proper invariant submodule.

## Orbits and stabilizers Edit

Consider a group G acting on a set X. The orbit of an element x in X is the set of elements in X to which x can be moved by the elements of G. The orbit of x is denoted by G.x:

$G.x = \left\{ g.x \mid g \in G \right\}.$

The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X. The associated equivalence relation is defined by saying xy if and only if there exists a g in G with g.x = y. The orbits are then the equivalence classes under this relation; two elements x and y are equivalent if and only if their orbits are the same; i.e., G.x = G.y.

The group action is transitive if and only if it has only one orbit, i.e., if there exists x in X with G.x = X. This is the case if and only if G.x = X for all x in X.

The set of all orbits of X under the action of G is written as X/G (or, less frequently: G\X), and is called the quotient of the action. In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.

### Invariant subsetsEdit

If Y is a subset of X, we write GY for the set g.y : yY and gG. We call the subset Y invariant under G if G.Y = Y (which is equivalent to G.YY). In that case, G also operates on Y by restricting the action to Y. The subset Y is called fixed under G if g.y = y for all g in G and all y in Y. Every subset that is fixed under G is also invariant under G, but not vice versa.

Every orbit is an invariant subset of X on which G acts transitively. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit.

A G-invariant element of X is xX such that g.x = x for all |gG. The set of all such x is denoted XG and called the G-invariants of X. When X is a G-module, XG is the zeroth group cohomology group of G with coefficients in X, and the higher cohomology groups are the derived functors of the functor of G-invariants.

## ΥποσημειώσειςEdit

1. Eie & Chang (2010). [[[:Πρότυπο:Google books]] A Course on Abstract Algebra]. σελ. 144.
2. This is done, e.g., by Smith (2008). [[[:Πρότυπο:Google books]] Introduction to abstract algebra]. σελ. 253.
3. 3,0 3,1 Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes, σελ. 175
4. tom Dieck, Tammo (1987), Transformation groups, de Gruyter Studies in Mathematics, 8, Berlin: Walter de Gruyter & Co., σελ. 29, ISBN 978-3-11-009745-0

## ΙστογραφίαEdit

Κίνδυνοι Χρήσης

Αν και θα βρείτε εξακριβωμένες πληροφορίες
σε αυτήν την εγκυκλοπαίδεια
ωστόσο, παρακαλούμε να λάβετε σοβαρά υπ' όψη ότι
η "Sciencepedia" δεν μπορεί να εγγυηθεί, από καμιά άποψη,
την εγκυρότητα των πληροφοριών που περιλαμβάνει.

"Οι πληροφορίες αυτές μπορεί πρόσφατα
να έχουν αλλοιωθεί, βανδαλισθεί ή μεταβληθεί από κάποιο άτομο,
η άποψη του οποίου δεν συνάδει με το "επίπεδο γνώσης"
του ιδιαίτερου γνωστικού τομέα που σας ενδιαφέρει."

Πρέπει να λάβετε υπ' όψη ότι
όλα τα άρθρα μπορεί να είναι ακριβή, γενικώς,
και για μακρά χρονική περίοδο,
αλλά να υποστούν κάποιο βανδαλισμό ή ακατάλληλη επεξεργασία,
ελάχιστο χρονικό διάστημα, πριν τα δείτε.

Επίσης,
(όχι μόνον, της Sciencepedia
αλλά και κάθε διαδικτυακού ιστότοπου (ή αλλιώς site)),
αν και άκρως απαραίτητοι,
είναι αδύνατον να ελεγχθούν
(λόγω της ρευστής φύσης του Web),
και επομένως είναι ενδεχόμενο να οδηγήσουν
σε παραπλανητικό, κακόβουλο ή άσεμνο περιεχόμενο.
Ο αναγνώστης πρέπει να είναι
εξαιρετικά προσεκτικός όταν τους χρησιμοποιεί.

- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν

>>Διαμαρτυρία προς την wikia<<

- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)


text:
mersenneforum.org Aliquot cycles search, the state of the art.
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 2015-05-23, 14:29 #1 Drdmitry     Nov 2011 23×3×11 Posts Aliquot cycles search, the state of the art. Here I collect all the information about the exhaustive search of small Aliquot cycles I know of. Amicable pairs. All pairs below 1e17 are known. Sergei Chernykh is currently running an exhaustive search of all 17-digit pairs. The range after 1e17 is relatively untouched. The most extensive list of all pairs can be found here: http://sech.me/ap/ Sociable numbers. We consider cycles of odd and even numbers separately. It is more efficient to look for odd cycles by their smallest elements because such Aliquot sequences tend to decrease. Even cycles are usually searched by the number preceding the largest number of the cycle. Odd cycles: All cycles below 1e15 are known. The search of all cycles between 1e15 and 1e16 is carried out by this forum here. Only cycles of a form given by Borho are known for higher numbers. Even cycles: All cycles below 5e12 are known for sure. It seems to me that Andre Needham completed the search for the range up to 1e13, but I am not 100% sure. Ed Hall started the search of all cycles of length at most four. Later he extended the length to ten. Currently he reached 5e13 (probably a bit more at the moment) for the cycles of length four. Only cycles of a form given by Borho are known for higher numbers. Potentially there may be a couple of longer cycles between 5e12 and 5e13. Theoretically there may be cycles containing both even and odd numbers. In this case they must contain a number of a form $2^d\cdot n^2$. A couple of years ago I searched for cycles containing a number of this form. Nothing was found up to 1e14. If there is an interest in doing a collective project to extend one of the ranges above then I can try to adopt my programs for it over summer vacations. Last fiddled with by Drdmitry on 2016-03-01 at 11:16 Reason: Updating
 2015-05-23, 16:47 #2 firejuggler     "Vincent" Apr 2010 Over the rainbow 3×5×179 Posts you mightwantto check http://mersenneforum.org/showthread.php?t=19372 edit : Ignore theling, Ididn't see you were already a part of this thread. edit 2,: maybe this one ? http://mersenneforum.org/showthread.php?t=13291 Last fiddled with by firejuggler on 2015-05-23 at 16:54
2015-05-23, 17:38   #3
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

838410 Posts

Quote:
 Originally Posted by firejuggler you mightwantto check http://mersenneforum.org/showthread.php?t=19372 edit : Ignore theling, Ididn't see you were already a part of this thread. edit 2,: maybe this one ? http://mersenneforum.org/showthread.php?t=13291
they've been a part of both at some point: http://mersenneforum.org/search.php?searchid=1630877

Last fiddled with by science_man_88 on 2015-05-23 at 17:39

2015-05-23, 17:58   #4
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3×29×83 Posts

Quote:
 Originally Posted by firejuggler you mightwantto check http://mersenneforum.org/showthread.php?t=19372 edit : Ignore theling, Ididn't see you were already a part of this thread. edit 2,: maybe this one ? http://mersenneforum.org/showthread.php?t=13291
This is a consolidated sequel to those threads.

2015-06-10, 12:00   #5
AndrewWalker

Mar 2015
Australia

2×41 Posts

Quote:
 Originally Posted by Drdmitry Here I collect all the information about the exhaustive search of small Aliquot cycles I know of. Amicable pairs. All pairs below 1e14 are known. Several people are searching pairs in a range between 1e14 and 1e15:Andrew Walker is doing a run with the 15d pairs where the first number has a highest factor from 10m to 40m. He is going to finish in a couple of months.
This is done and sent to Patrick, along with a quick run on 16 digits up to 10000 for factors of the first number. He is still going
through them, but for the 15 digits so far only <5% are new. I'll be doing more now with 16 digits and some of the breeding methods which look promising!
Andrew

2015-06-11, 10:23   #6
Drdmitry

Nov 2011

23×3×11 Posts

Quote:
 Originally Posted by AndrewWalker This is done and sent to Patrick, along with a quick run on 16 digits up to 10000 for factors of the first number. He is still going through them, but for the 15 digits so far only <5% are new. I'll be doing more now with 16 digits and some of the breeding methods which look promising! Andrew
That is great! I am afraid that I do not have the rights to change the initial text so it will remain unchanged unfortunately.

As far as I understand, your methods are specific for finding amicable pairs and can not be used for longer cycles? Currently I am interesting in cycles of length bigger than two. My programs are not fast enough to search over all even 15 digits numbers or all odd 16 digits numbers on one computer in a reasonable time. However it can be done with help of a hundred cores.

2015-06-14, 14:47   #7
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

22×33×5×11 Posts

Quote:
 Originally Posted by Drdmitry That is great! I am afraid that I do not have the rights to change the initial text so it will remain unchanged unfortunately. As far as I understand, your methods are specific for finding amicable pairs and can not be used for longer cycles? Currently I am interesting in cycles of length bigger than two. My programs are not fast enough to search over all even 15 digits numbers or all odd 16 digits numbers on one computer in a reasonable time. However it can be done with help of a hundred cores.
PM me/Post changes and I will add them.

2015-06-15, 12:33   #8
AndrewWalker

Mar 2015
Australia

5216 Posts

Quote:
 Originally Posted by Drdmitry That is great! I am afraid that I do not have the rights to change the initial text so it will remain unchanged unfortunately. As far as I understand, your methods are specific for finding amicable pairs and can not be used for longer cycles? Currently I am interesting in cycles of length bigger than two. My programs are not fast enough to search over all even 15 digits numbers or all odd 16 digits numbers on one computer in a reasonable time. However it can be done with help of a hundred cores.
That's correct, it has bits in it which cut off a lot of possible inputs when it determines they can't be an amicable pair, so it
would miss a lot of larger cycles. Wouldn't we all like to have a hundred cores to play with!

Andrew

2015-06-15, 16:18   #9
Drdmitry

Nov 2011

23·3·11 Posts

Quote:
 Originally Posted by AndrewWalker That's correct, it has bits in it which cut off a lot of possible inputs when it determines they can't be an amicable pair, so it would miss a lot of larger cycles. Wouldn't we all like to have a hundred cores to play with! Andrew
At the moment I am adjusting the program to make it comfortable for distributed computing.
Unfortunately the code is 64 bit Windows specific. However the only specific part is a fast 64 bit multiplication modulo a 64 bit number. Hopefully it is not too difficult to rewrite it on any other 64 bit system.

 2015-06-30, 08:22 #10 AndrewWalker     Mar 2015 Australia 2·41 Posts All of my Walker&Einstein results from above have been processed by Pat and are in his lists. There were over 40 thousand submitted, the number of new pairs are 15 digits 656 pairs 16 digits 18018 pairs I'm mainly interested in finding new type n1 or Xn1 pairs, the two new ones were X31 Walker&Einstein 2015 1691429789009025=3^5*5^2*13*71*401*709*1061 1756294497513855=3^5*5*13*71*1566099539 X41 Walker&Einstein 2015 5804063919860661=3^2*7^2*13*19*29*47*89*101*4349 6126766409739339=3^2*7*13^3*19*29*80335799 I'm also hoping to find new pairs where the smallest prime factor is bigger than 3. Can you please add above I'm looking at even 16 digits with the largest factor from 10k to 100k in the first number, this won't find everything and will take several months! Andrew Last fiddled with by AndrewWalker on 2015-06-30 at 08:23
2015-06-30, 09:36   #11
Drdmitry

Nov 2011

4108 Posts

Quote:
 Originally Posted by AndrewWalker All of my Walker&Einstein results from above have been processed by Pat and are in his lists. There were over 40 thousand submitted, the number of new pairs are 15 digits 656 pairs 16 digits 18018 pairs I'm mainly interested in finding new type n1 or Xn1 pairs, the two new ones were X31 Walker&Einstein 2015 1691429789009025=3^5*5^2*13*71*401*709*1061 1756294497513855=3^5*5*13*71*1566099539 X41 Walker&Einstein 2015 5804063919860661=3^2*7^2*13*19*29*47*89*101*4349 6126766409739339=3^2*7*13^3*19*29*80335799 I'm also hoping to find new pairs where the smallest prime factor is bigger than 3. Can you please add above I'm looking at even 16 digits with the largest factor from 10k to 100k in the first number, this won't find everything and will take several months! Andrew
I'm afraid I do not have rights to edit posts here. You can send a private message to henryzz and ask him to make make the changes to the post. I can also do that but the first variant seems to me more efficient.

Some time ago I did the search of all 16d Aliquot cycles with the smallest factor bigger than 3. There is only one such cycle already found by you. I did not check bigger numbers though.

 Similar Threads Thread Thread Starter Forum Replies Last Post Drdmitry Aliquot Sequences 25 2016-12-16 15:26 Drdmitry Aliquot Sequences 302 2016-05-11 02:17 schickel Aliquot Sequences 7 2013-02-08 01:33 Drdmitry Aliquot Sequences 0 2011-12-14 13:50 R. Gerbicz Math 0 2010-07-01 12:30

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GR 8677927796770177 | # Login | Register

GR8677 #68
Problem
 GREPhysics.NET Official Solution Alternate Solutions

Special Relativity$\Rightarrow$}Relativistic Energy

The relativistic energy relation is,

$E= \sqrt{p^2 c^2 + mc^4.}$

As the speed of a particle $v \rightarrow c$, $E\rightarrow pc$, since by the de Broglie relation, $p=h/\lambda=h\nu/c=E/c$. Plug in the requirement into the energy relation to find that $m\rightarrow 0$, as in choice (A).

Alternate Solutions
 pranav2012-11-09 13:17:09 One more way of doing this is by recalling the following equation: $m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$ The only way anyone (including Mr. Photon) can go at $v=c$ is by having a rest mass of zero. (For those who doubt this: The above equation can be re-written at v=c as: $m_0=m \sqrt{1-\frac{v^2}{c^2}}=m (1-1)=m (0)=0$ QED) Reply to this comment
Giubenez
2014-10-17 05:46:11
The easiest way is surely to compare it with the photon (that is known to have a rest mass = 0).
Since
$p=\gamma \cdot m \cdot v$
and we know that photons have finite momentum the only possible explanation is that
$m_\gamma = 0$
($\gamma \Rightarrow \infty$).
Now our particle, having v =c must have m=0 to obtain a finite momentum..
pranav
2012-11-09 13:17:09
One more way of doing this is by recalling the following equation:

$m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

The only way anyone (including Mr. Photon) can go at $v=c$ is by having a rest mass of zero.

(For those who doubt this:

The above equation can be re-written at v=c as:

$m_0=m \sqrt{1-\frac{v^2}{c^2}}=m (1-1)=m (0)=0$

QED)

Astrophysicist
2011-10-02 11:34:45
One can also remember that photons(which travel at the speed of light) have rest mass = 0. Thus, this particle must have a rest mass = 0 as well.
 mpdude82012-04-19 15:15:23 You can't really say "particle X will automatically share the same characteristics of a photon" and deduce the answer from there, otherwise answer B also would be valid.
shak
2010-07-31 21:20:43
Can we make a conclusion from this formula,
m=m(0)/(1-v^2/c^2)^0.5
then,
m(o)=m*(1-v^2/c^2)^0.5

when v approaches c, then m(0) goes to zero
Is this right?
thanks
 neon372010-11-03 11:55:06 umm....that looks correct to me. I cant think of an arguement to counter that (atleast yet).
spacemanERAU
2009-10-20 08:37:41
Also, this problem is really simple if you remember that any particle that has speed of c MUST have rest mass of zero...no math involved
2006-10-23 00:41:07
It should be $E=\sqrt{p^2c^2+m^2c^4}$
 spacemanERAU2009-10-18 18:35:28 agreed!

LaTeX syntax supported through dollar sign wrappers $, ex.,$\alpha^2_0$produces $\alpha^2_0$. type this... to get...$\int_0^\infty$$\int_0^\infty$$\partial$$\partial$$\Rightarrow$$\Rightarrow$$\ddot{x},\dot{x}$$\ddot{x},\dot{x}$$\sqrt{z}$$\sqrt{z}$$\langle my \rangle$$\langle my \rangle$$\left( abacadabra \right)_{me}$$\left( abacadabra \right)_{me}$$\vec{E}$$\vec{E}$$\frac{a}{b}\$ $\frac{a}{b}$


text:
The common variants are isosceles, equilateral, scalene etc. $\displaystyle \widehat{A}=\widehat{E}$ ; $\displaystyle \widehat{B}=\widehat{F}$. SSS – Side Side Side Rule for Triangles We can Triangles, of course, have their own formulas for finding area and their own principles, presented here: Triangles also are the subject of a theorem, aside from the Pythagorean one mentioned earlier. The AAS Rule (two Angles and a corresponding Side) for showing that two triangles must be congruent, with a demonstration why the side must … When we have proved the two triangles in congruence through this benchmark, the remaining two sides and the third angle will also be equal. = as opposite sides of parallelogram are equal in length. This gives another rule which lets you see if two triangles are congruent. SAS Congruence Rule (Side – Angle – Side) ABC = ADC. Then, the riangles ABC and EFG are congruent. Two triangles are congruent if all their corresponding angles have the same measure and all their corresponding sides have the same length. Pro Subscription, JEE Leave out any A that stands for a right angle. Hence, there is no AAA Criterion for Congruence. There are FOUR “Shortcut Rules” for Congruent Triangles that we will be covering in this lesson. Congruent Triangles Triangles are the most primary shapes we learn. Similarly for the angles marked with two arcs. Application of congruent triangles into architecture has a good valid reason. 2. 2. Imagine of all the pawns on a chessboard and they are congruent. Easiest Way to Find if the Triangle is Congruent, By this rule, two triangles are congruent to each other - If one pair of corresponding sides and either of the two pairs of angles are equivalent to each other. From the above diagram of three triangles, you can observe that given triangle XYZ can be any of the following and we are not sure which diagram of Triangle ABC is congruent to Triangle XYZ. Four rules of proving that two triangles are congruent Rule 1 : The SSS rule: Side-Side-Side rule The side-side-side rule states that if the three sides of a triangle are equal to the three sides of the other triangle then those two triangles are congruent. In simple terms, any object when laid over its other counterpart, appears to be the same figure or Xerox copies of each other are congruent. In the above figure, Δ ABC and Δ PQR are congruent triangles. Two bangles of the same shape and size are congruent with each other. Thus, two triangles can be superimposed side to side and angle to angle. is a parallelogram. What are the Real Life Applications of Congruent Triangles? Similar triangles - Higher Two triangles are similar if the angles are the same size or the corresponding sides are in the same ratio. In the simple case above, the two triangles ABC and DEF are congruent as each of their corresponding sides are equal, and all corresponding interior angles have the same measure. There are a number of pairs of triangles that are used in structuring buildings. By this property a triangle declares congruence with each other - If two sides and the involved interior angle of one triangle is equivalent to the sides and involved angle of the other triangle. The property is based on making a triangle congruent depending on how many sides and angles of equal measures make a congruent pair. The criteria for congruence of triangles class 9 is explained using two axiom rules. There is also another rule for right triangles called the Hypotenuse Leg rule. It is called the Angle-Side-Angle or ASA rule for congruence of triangles. Oct 1, 2018 - Teacher's Math Resources blog - a collection of free and paid resources for teachers. What’s amazing is that no matter how you keep flipping it, the other triangle i.e “DEF” will rotate to remain in congruence to triangle “ABC” and vice-e-versa. As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent. When two triangles are congruent we often mark corresponding sides and angles like this:The sides marked with one line are equal in length. It’s called the SSS rule, SAS rule, ASA Two triangles with equal corresponding angles may not be congruent to each other because one triangle might be an enlarged copy of the other. So, what are congruent triangles? Side-Angle-Sideis a rule used to prove whether a given set of triangles are congruent. That’s why based on the  the side – angle – side rule states that if two sides and the angle between those two sides are equal to the two sides and the angle between them of the other triangle, then those two triangles are congruent. This means, Vertices: A and P, … The common variants are equilateral , isosceles, scalene Then the triangles ABC and EFG are congruent ABC = EFG. In this case, two triangles are congruent if two sides and one included angle in a given triangle are equal to the corresponding two sides and one included angle in another triangle. The SAS rule states that If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent. Rules that do not Apply to Make Congruent Triangle, Vedantu Then, the riangles ABC and EFG are congruent, ABC = EFG, Rule 3: The AAS rule: Angle – Angle – Side rule. Thus, the Triangles will be congruent based on certain properties that are as follows. We recall that this is the angle – side – angle rule states that if one side and the two angles sideways this side of the triangle are equal to the side and the two angles sideways this side of the other triangle then those triangles are congruent. The first of these “Shortcut Rules” is the “Side Side Side”, or “SSS” Rule. Corresponding Parts In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. By this rule, two triangles are congruent to each other - If two angles and the involved side of one triangle is equivalent to the two angles and the included side of the other triangle. There are four rules to check for congruent triangles. Pro Lite, NEET The angle at “B” measures the same (in degrees) as the angle at “E”, while the side “BA” is the same length as the side “ED” etc. 1. This specific congruent triangles rule represents that if the angle of one triangle measures equal to the corresponding angle of another triangle, while the lengths of the sides are in proportion, then the triangles are said to have passed the congruence triangle test by way of SAS. In our case we have two corresponding internal angles that are equal with each other. Two triangles are said to be congruent if all 3 3 of their angles and all 3 3 of their sides are equal. As closed figures with three-sides, triangles are of different types depending on their sides and angles . The congruence of triangle enables the architect to compute the forces exerted on the building, thus ensuring that the forces are in equilibrium, ultimately that the building will not fall flat. Using : is common. Every triangle is typically represented by 6 measures i.e. = for same reason. Congruent triangles cannot be expanded or contracted, and still be congruent. Also in how far doors swing open. If EF is greater than EG, the diagram below shows how it is possible for to "swing" to either side of point G, creating two non-congruent triangles using SSA. So, $\displaystyle \Delta$ABC and $\displaystyle \Delta$ADC are congruent. Rule 4: The ASA rule: Angle – Side – Angle rule. $\displaystyle \widehat{B}=\widehat{F}$ ; $\displaystyle \widehat{C}=\widehat{G}$. Prove that triangles and are congruent. Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. For example, congruent triangles are executed into the design of roof ends, such that the beam of the roof and the uppermost edges of the walls are horizontal. Amongst various others, SAS makes for a valid test to solve the congruent triangle problem. When we look into this two triangles ABC and ADC we found that we have two corresponding angles that are equal. This rule is a self-evident truth and does not need any validation to support the principle. The criterion of this principle is the Angle sum property of triangles that suggests that the sum of 3 angles in a triangle is 180°. And what I want to do in this video is figure out which of these triangles are congruent to which other of these triangles. This is the first criterion for congruence of triangles. Thus, we can say that they are congruent. These two triangles are of the same size and shape. There are a variety of tests conducted to find the congruence between two triangles. Moreover, pairs of triangles are used especially in situations where it is beyond one's capability to physically calculate the distances and heights with normal measuring instruments. The angle-angle-side rule states that if two angles and one of the side in front of one of the angles of the triangle are equal to the two angles and the other side of the other triangle then those two triangles are congruent. We also see that the diagonal of the parallelogram is a common side to both of our triangles. There are 5 rules through which we can prove that two triangle are congruent or not: 1) SSS-means SIDE-SIDE-SIDE i.e, if two triangles have all three sides equal they are then congruent. We already saw two triangles above, but they were both congruent. $\displaystyle \widehat{ADB}=\widehat{CDE}$ because they are opposite angles. ∴ Triangles and … $\displaystyle \left[ AD \right]=\left[ DE \right]$, Because the point D is the middle point of the segment $\displaystyle \left[ AE \right]$, 2. Why are Congruent Triangles Put into Architecture? • If two triangles ABC and PQR are congruent under the correspondence P,B Q and then symbolically, it is expressed as Worked Example 2: The segments  $\displaystyle \left[ AE \right]$ and $\displaystyle \left[ BC \right]$ intersect in the point D, which is the middle point of each of this segments. When we have proved the two triangles in congruence through this benchmark, the remaining two sides and the third angle will also be equal. Congruent Triangles two triangles are congruent if and only if one of them can be made to superpose on the other, so as to cover it exactly. Worked example 1: We are given the parallelogram ABCD. The side-side-side rule states that if the three sides of a triangle are equal to the three sides of the other triangle then those two triangles are congruent. Do you know cigarettes in a packing are in congruence to each other. So, $\displaystyle \Delta$ABC and $\displaystyle \Delta$ CED are congruent. In a similar vein, different various groups of three will do the needful. Then the triangles ABC and EFG are congruent, Prove that the diagonal AC divides the parallelogram in two congruent triangles. 3 sides & three angles. Hence, there is no AAA Criterion for Congruence. 4 2 triangle congruence by sss and sas pdf 5 Using Congruent Triangles 4. Under this criterion of congruence— when two equal sides and one equal angle forms the two similar sides, it will result in triangles appearing similar. 3. Similarly for the sides marked with two lines. In the diagram of AABC and ADEP below, AB z DE, ZA ZD, and LB z ZE. It can be told whether two triangles are congruent without testing all the sides and all the angles of the two triangles. Main & Advanced Repeaters, Vedantu In fact, any two triangles that have the same three side lengths are congruent. Although these are 6 6 parameters, we only need 3 3 to prove congruency. The application of triangles identical in shape and size is of utmost significance, because of the gravitational property of the congruent triangles. Based on the properties of the parallelogram we know that the opposite sides are parallel and congruent. It will be a case of Two triangles of the same shape, but one is bigger than the other. The three-angled, two-dimensional pyramids known as triangles are one of the building blocks of geometry (however three-cornered they may be). Rules for Two Triangles to be Congruent Rule 1 : SSS (Side, Side, Side) Two triangles can be congruent, if all the three sides of a triangle are equal to the corresponding sides of … SSS, SAS, ASA, AAS, and HL...all the … An included angleis an angle formed by two given sides. 3. Find the AB, if CE = 10 cm. Thus, if two triangles are of the same measure, automatically the 3rd side is also equal, therefore forming triangles ideally congruent. The angle-side-angle rule states that if one side and the two angles sideways this side of the triangle are equal to the side and the two angles sideways this side of the other triangle then those triangles are congruent. The side-angle-side rule states that if two sides and the angle between those two sides are equal to the two sides and the angle between them of the other triangle then those two triangles are congruent. If the side which lies on one ray of the angle is longer than the other side, and the other side is greater than the minimum distance needed to create a triangle, the two triangles will not necessarily be congruent. Activities, worksheets, projects, notes, fun ideas, and so much more! Solution: Based on the properties of the parallelogram we know that the opposite sides are parallel and congruent. Nov 25, 2016 - Everything you ever needed to teach Congruent Triangles! Then, the riangles ABC and EFG are congruent, ABC = EFG, Rule 2: The SAS rule: Side – Angle – Side rule. They are called the SSS rule, SAS rule, ASA rule and AAS rule. If two sides and an included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. The segments  $\displaystyle \left[ AE \right]$ and $\displaystyle \left[ BC \right]$ intersect in the point D. which is the middle point of each of this segments. Then the triangles ABC and EFG are congruent, ABC = EFG. By this rule of congruence, in two triangles at right angles - If the hypotenuse and one side of a triangle measures the same as the hypotenuse and one side of the other triangle, then the pair of two triangles are congruent with each other. $\displaystyle \widehat{BCA}=\widehat{CAD}$, $\displaystyle \widehat{BAC}=\widehat{ACD}$. Congruent Triangles Definition: Triangles are congruent when all corresponding sides and interior angles are congruent.The triangles will have the same shape and size, but one may be a mirror image of the other. The criterion of this principle is the Angle sum property of triangles that suggests that the sum of 3 angles in a triangle is 180°. But the fact is you need not know all of them to prove that two triangles are congruent with each other. The Altitude-on-Hypotenuse Theorem makes […] Though the triangles will have the same shape and size, one will appear as a mirror image of the other. Find the AB, if CE = 10 cm. Two triangles are said to be congruent if their sides have the same length and angles have same measure. The congruent triangle is certainly one of the appropriate ways of proving that the triangles are similar to each other in both shape and size. As a plane enclosed figures with 3-sides, segments - “triangles” are of different types based upon their sides and angles. And since we can be sure the triangles are congruent, this suggests that the three angles of one triangle are equal to the angles of the other triangle respectively. Hence, this confirms that two triangles cannot be congruent, if one side of a triangle is equal to the corresponding side of another triangle. For two triangles to be congruent, one of 4 criteria need to be met. A surprising phenomenon of congruent triangles as well as other congruent shapes is that they can be reflected, flipped or converted , and still remain congruent. Welcome to Clip from. $\displaystyle \left[ BD \right]=\left[ DC \right]$, Because the point D is the middle point of the segment. SSS Congruence Rule (Side – Side – Side) Two triangles are said to be congruent if all the sides of a triangle are equal to all the corresponding sides of another triangle. Solution: If we see the figure we have that: 1. Thus, if two triangles are of the same measure, automatically the 3. side is also equal, therefore forming triangles ideally congruent. This is called the SSS Congruence Condition for triangles (“Side-Side-Side”). By this rule, two triangles are congruent to each other - If two angles and the involved side of one triangle is equivalent to the two angles and the included side of the other triangle. Sorry!, This page is not available for now to bookmark. Repeaters, Vedantu So the two original triangles are congruent. Also for the sides marked with three lines.The angles marked with one arc are equal in size. We also know that when two parallel lines are intersected by a third one we know that the alternate internal angles have equal measures, also the alternate external angles have equal measures. Can we say SAS is a Valid Similarity Theorem? From this we have that AB = CE, which means that AB = 10 cm. Axiom 7.1 (SAS congruence rule) :Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle. Also for the angles marked with three arcs. SSS stands for \"side, side, side\" and means that we have two triangles with all three sides equal.For example:(See Solving SSS Triangles to find out more) By this rule, two triangles are congruent to each other - If one pair of corresponding sides and either of the two pairs of angles are equivalent to each other. Now that all three corresponding sides are of the same length, you can be confident the triangles are congruent. By this rule, two triangles are said to be congruent to each - If all the three sides of one triangle are of same length as all the three sides of the other triangle. By this rule, if all the corresponding angles of a triangle measure equal, the triangles will become about the same shape, but not necessarily the same size. Prove that the diagonal AC divides the parallelogram in two congruent triangles. What we have drawn over here is five different triangles. Four rules of proving that two triangles are congruent. Given two sides and a non-involved angle, it is likely to form two different triangles that convince the values, but certainly not adequate to show congruence. In this lesson, we'll consider the four rules to prove triangle congruence. AABC = A DEF 5 Do you need all six ? So, we have one equal side and the two angles sideways the side that are equal. In congruent triangles in front of congruent angles $\displaystyle \widehat{ADB}=\widehat{CDE}$, There are congruent side lengths $\displaystyle \left[ AB \right]=\left[ CE \right]$. Side – Angle – Side Side Angle Side (SAS) is a rule used to prove whether a given set of triangles are congruent. Pro Lite, Vedantu Vedantu academic counsellor will be calling you shortly for your Online Counselling session. ABC = ADC. Triangles are said to be in congruence when every corresponding side and interior angles are congruent (of same length). Triangles ABC and $\displaystyle \Delta$ ABC and EFG are congruent triangles,! 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congruent triangles rules 2021


text:
8:04 AM
LaTeX2e <2021-11-15> patch level 1

8:44 AM
@JosephWright oh I was expecting siunitx

@JosephWright do we get also an accompanying dev release?

(from the look of it) It really simulates TeX?
... wouldn't it be better to just modify TeX source code then...

9:03 AM
@PhelypeOleinik I'll try to wrap my head around that.
@PhelypeOleinik For the record, some or all of that explanation should be in the answer itself. Typically answers on TeX SE contain too much of "here is how you do it", and not enough of "here is what this means", and "here is how this works".
Magical recipes are fine in Harry Potter, but aren't that useful in real life.

@PauloCereda :)
@UlrikeFischer At some stage I guess: I was not sure it was super-urgent
@user202729 Yes, that's the plan
@user202729 Needs a different skillset, means getting engines built (and named), ...
@FaheemMitha The usual tension that in the end 'read the manual' covers much of 'how does it work' (certainly 'read the sources' does, as we can go all the way to tex.web)

@JosephWright I don't think we need to go that far. :-)

@FaheemMitha :)

But many answers contain no explanation at all. And much of the details of how TeX works aren't exactly intuitive for the non-expert.
I think there is room to strike a happy balance.

@FaheemMitha Oh, definitely
@FaheemMitha I tend toward a 'Here is what to do ---- Here is how it works' approach

9:11 AM
For example, @PhelypeOleinik's answer I just referenced isn't exactly staggeringly obvious, even if you have some idea what catcodes are, and what they do.

@FaheemMitha Ah, but the problem then becomes how much to detail

@JosephWright Indeed, hence "happy balance".

@UlrikeFischer Let me check my l3build change works

Phelype's answer contains some explanation, so I guess that was his happy balance.
But I would have been happier with more.

The problem here is...
It's really very hard to understand how these "fiddle with this internal command and this other internal command produces this result" works
The only way is to read their source code.
If you already understand catcode, that is.

9:16 AM
For example, he wrote a comment,which says in part:
> The catcode manipulation is because the varbatim-isation of filecontents makes the backslash be an ordinary character; thus you need something else.
Which seems like an important detail to me.
@user202729 I can believe that.
OTOH, perhaps his comment is obvious.
@user202729 But I'm making a general point. @PhelypeOleinik's answer has some explanation. It's quite common for answers to have no commentary whatsoever, not even to indicate general strategy.

... yes, I complained about the same thing not-very-recently.

I suppose one can take the viewpoint of - people need to figure it all out themselves. And I suppose that's a legitimate viewpoint in some sense.

I'm newer than you are, so I even have difficulty figuring out where the documentation is. (in the end I write a new answer for it.)

@user202729 I recommend "TeX by Topic", which is free and part of TeX Live.
It's not super comprehensive, but very helpful, and organized in a sensible way.

Yes, already figured out these and others.

9:20 AM
texdoc texbytopic
There's also that free book floating about on the net, which is occasionally helpful.
This one does have errors, though. But they fix the errors.
This manual has made a start at collecting the huge amount of LaTeX information available in one place. And which the printed books typically don't have space to include.
@user202729 Figured out what?

(That the books you mentioned exists.)
@JosephWright True that.
Named?
Although yes, on the other hand if you just have a patch to the source code it will over time get merge conflicts and stuff

10:04 AM
@user202729 but then you'd have to distribute the modified binaries, and have modified pdftex and luatex and xetex and .. and so it would be a different thing altogether.

10:19 AM
@user202729 As in 'different to pdftex, etc.'

10:29 AM
@HenriMenke I'm going to work on the test suite/l3build today, I think I see what I need to do

11:06 AM
Just post the patch and let interested readers figure out how to compile TeX themselves
Basically the "I want to share this with the world but too lazy to support it" kind

@user202729 Er, that would be a very small set of people (DEK, @DavidCarlisle and I when we have to, @PhelypeOleinik, the small set of people who provide TL binaries, Christian Schenk, that's probably it)

But then people have more motivation to figure out how to build from source
The set of people who build gcc from source is not just the gcc developers either

@user202729 Oh, the KerTeX guy
@user202729 Perhaps, but why would anyone want to?
@user202729 Somehow I doubt it ... it's not ultra-difficult, but it's not exactly a routine task either
@user202729 I guess I'd point out that nowadays even the LaTeX team don't bother building their own LaTeX formats, we just leave it to TeX Live :)

@user202729 the people who want to trace tex macro expansion and who need to debug compiling tex source are almost completely disjoint.

I didn't ever mentioned debug compiling TeX source though?

11:16 AM
@user202729 you suggested using a modified tex source above didn't you?
@user202729 compiling tex on a new machine isn't exactly unlikely to fail without some lowish level debugging of library dependencies
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Then hopefully someone will make it better, with cmake or conan or something
Not that I know how to use these things

@user202729 There's a make recipe ...

But?

@user202729 and you have cmake set up on windows, and cygwin and android and rasberry pi and all the other texlive platforms?

@user202729 Fine if you are using Linux, less fine for macOS or Windows (I do any engine-compilation tests on a Linux VM as a result)

11:19 AM
@user202729 not to mention the 7 million tex users using overleaf and no access to a commandline at all

@DavidCarlisle :)
@user202729 I guess really what I'm trying to get at is that for almost anyone debugging some TeX thing, asking to use a self-compiled engine is not going to help: they are much better off with \tracingall or similar. Using unravel is a way to lower the barrier from within a setup they already have

11:37 AM
@JosephWright Your efforts regarding l3draw are outstanding and the backends are well-tested, so it would be stupid to not make use of them. I expect that this will also solve all the problems we currently have with XeTeX.
2
expl3 also provides useful abstractions for common algorithms like walking lists and expanding arguments.
But I can also say for certain that pgfmath will never be replaced by l3fp.

@HenriMenke Sure: you can't do widthof or similar by expansion

11:52 AM
\fpeval is coming ...

By the way I made something to let 2^\sqrt 2 works without the extra braces. (why isn't it the default?)

@user202729 What do you mean 'work'? The input is ambiguous at best, and there's the whole 'active ^ with an argument' business

> What do you mean "work"?
In my case... at least that particular example outputs what you would expect (same as 2^{\sqrt 2} before the change)

@user202729 the markup is simply horrible though it is really unfortunate that 2^\mathrm x works without braces
@user202729 depending how you did it you probably broke tex4ht and most other tex-to-anything translators though.

Yes, I understand that currently they are not intentionally made to be error or not error

11:58 AM
@user202729 The other way round would be better: make 2^2 not work so users get into the habit to always use {} and thus avoid problems in the future :)

But that's really annoying to read

@samcarter That's quite easy to do

Hm? I thought that one is impossible
Unless you make something active
But the {}s are annoying to read

@user202729 don't use tex syntax if you don't like {}

@user202729 you get used to them very quickly. In the long run, the make it actually easier to read.

11:59 AM
After using it for a while I conclude that TeX has almost the best most flexible syntax, but not completely flexible
Compared to Python "have to put quotes almost everywhere" if you want to use it for the same purpose

@user202729 Exactly, make ^ active and do a look-ahead

@user202729 but actually I think x^2 is fine. It is unfortunate that the argument of ^ is not parsed the same way as the argument to a macro, so a single token or brace group. the fact that ^ expands as needed at the primitive level to find a brace group means that it has a syntax completely different from all other top level latex constriucts

Let's see...
The reason why there's the extra complexity in TeX the engine in the first place is to make it a little easier to read
(compare \^{2}{3})
Actually since ^ doesn't really take the left one as input (does it?)
Still, there's \over in plain TeX
Niceness and consistency should be balanced, I guess. Too much consistency leads to lisp's (+ 2 3)

@user202729 \over is absolutely the worst thing in the tex design and responsible for much of the complication with math fonts and \mathchoice weirdness.
2

And too much niceness leads to C++? Nearly impossible to parse, too many special cases (T::template f<int>(1), std::vector<std::vector<int>>)
Then in that respect, somehow it seems that TeX is neither nice nor consistent (let's say it's historical artifact then. You can't just throw it away and redesign the core)

12:11 PM
@user202729 sure (well you can, start again see Don H's finl for example finl.xyz but for most people most of the time working within tex constraints meets most requirements)

Yes, you pointed that one to me once before
Last time I remember it being incomplete or idea-only
Still, with roughly one developer and about one year that's... okay I guess

@user202729 idea only (mostly) as far as I know.

So in other words "not usable at the moment"; still, idea is good (only) if I'm willing to tackle something that large
Back to the ^\sqrt thing, so the consensus is mostly "allow nothing but ^<digit>"?
Why not be more flexible? User can choose to use all-{} or not
Or an opt-in library (... it's just a library, why don't I make one myself?)

12:33 PM
Okay looks like parsing \tracingall output programmatically is hard
After \escapechar is changed it also changes in output?
By the way do anyone know anything that compiles Python to expl3, or something similar?

12:55 PM
@user202729 say what?!

@user202729 Why would one do that?
@user202729 Yes, \tracingall respects \escapechar

@JosephWright Because...
It took me way too long to get used to writing expl3 syntax
Look, someone did write PyLaTeX
Was trying to look for ways to parse tracingall output but looks a little... hard

Programmically.

@user202729 Yes, but that's at the product-a-document level, whereas expl3 is TeX programming so has rather different requirements
@user202729 See my previous answer ;) (As in, if I'm stuck I just work through on paper imagining I'm TeX)

1:00 PM
Still, I guess the point is that someone find TeX too hard to read

@user202729 some things in life come at a cost. :)

Both writing something-else and put up with the current situation is hard.

@user202729 People don't read \tracingall because it's fun, but because there's a (usually well hidden) bug. Making a buggy parser for that would hardly improve things
2

@PhelypeOleinik You sure that @DavidCarlisle doesn't consider it a fun pastime while waiting for the duck to get crispy?

@samcarter oh no

1:12 PM
@samcarter Well, maybe @DavidCarlisle does it for fun, because he doesn't know how to make bugs :)

@PhelypeOleinik :) must be such a boring life without bugs

@samcarter Indeed :)

@samcarter /entomologists in the audience just fainted

My point is...

@PauloCereda <3

1:22 PM
I'm thinking a non-buggy parser is possible.
But because it's full of special cases it turns out to be hard...
There obviously isn't any way to tell TeX to digest the next token (\digestafter like \expandafter) right? (actually there might be with LuaTeX. Should look at it later...)
If it were that easy unravel wouldn't need to do such hard things

/Gödel intensifies

2:16 PM
@DavidCarlisle -- I queried DEK specifically about this. His response was essentially that his goal was that TeX input should sound natural to a mathematician when spoken. \frac doesn't, even though it's easier to parse. This is a key place where one recognizes his origin as a mathematician rather than as a computer scientist.

@user202729 This will break when the command uses runtoks internally, but otherwise you can use
\directlua{
tex.runtoks(function()
token.scan_token()
tex.quittoks()
end)
print'This comes between the two \noexpand\\show commands'
}\show\abc\show\ABC
\bye
Actually it might not even break then. Probably does weird things when nested though.
@PauloCereda Don't tempt me. I don't have time to obsess about my neovim setup for the rest of the day... :)

@barbarabeeton yes I have seen his comments on over before. Even great men can be wrong sometimes:-)

@MarcelKrüger <3

@user202729 no don't special case digit. If I had a choice I would do (what tex4ht does) and make ^ active and defined to take a normal #1 argument (so a single token or brace group)
@user202729 flexibility is good in places but having to run the full tex execution mechanism in order to be able to parse the argument to a superscript command means there is no possibility for any sort of grammar or any reliable tex-to-anything translation that doesn't use tex

2:34 PM
@PauloCereda <3

@user202729 is it reasonable to expect editors to correctly syntax highlight this for example
\def\foo{\bgroup  abc}

$x^\foo xyz} + 5$

\bye

2:51 PM
@DavidCarlisle \@backslashchar broke xcolor?

@UlrikeFischer Oh no!

@UlrikeFischer the question that you just commented on?

@DavidCarlisle if you mean the bug report in xcolor.

@UlrikeFischer oh no haven't seen that (I'm at Mum's and only online occasionally)
@UlrikeFischer ah yes sorry I did see it your "comment" was in the the issue email, OK, that's due to the backslashchar change?

@DavidCarlisle I think yes, that's the place where the tracing starts to differ.

3:05 PM
@UlrikeFischer I guess I should look...

@DavidCarlisle this here has changed too. Now you get relax:
\documentclass{article}
\usepackage{color}

\begin{document}
\expandafter\show\csname\string\color@red\endcsname
\end{document}

(The original one in plain TeX was \over. Its \frac was different.)
So it's fair to say that plain TeX (tries to) maximize readability while keeps implementation/reading cost reasonable...
(over time it becomes unreasonable.)
I'm talking about the specific case 2^\sqrt 2 here
Yes, of course, as long as it's possible to redefine catcode in TeX it's impossible to syntax highlight correctly in general (that one doesn't even redefine catcode)
Thinking about it, how easy/hard would it be to correctly parse 2^\sqrt 2 without a TeX engine? Actually pretty hard
(compare 2^\alpha)

3:58 PM
@user202729 Sure you could throw it away and resign the core. It would just be a tremendous amount of work.
And most people would probably do something different, anyway. They aren't necessarily enamored of TeX's design.
show -> shows.

@user202729 it's the same as my \foo example above, if it works at all then you need to know \sqrt inserts a bgroup at the start and need to know what tokens it absorbs and where it inserts an egroup to end the superscript.
@UlrikeFischer that would be wrong but it's mum's birthday and we are having duck a cup of tea so color will have to wait

4:33 PM
@DavidCarlisle phew tea

@PauloCereda it is possible I made an error in the markdown and struck out the wrong phrase

@DavidCarlisle oh no

@DavidCarlisle I think the problem is a spurious space ;-) \csname\string\color @#1\endcsname eats the space after "color", but \csname\@backslashchar color @#1\endcsname doesn't.

@PauloCereda ohh

4:59 PM
@UlrikeFischer grr I thought I'd caught those: better make an update I guess but I'll be driving home shortly so will be later tonight unless you beat me to it:-)

@DavidCarlisle xcolor seems ok, only color is wrong. I can make a pull request and you can then check, I will be away in the evening too.

@UlrikeFischer thanks

@DavidCarlisle thinking about the commit message ...

@UlrikeFischer Get @egreg to do it ;)

@UlrikeFischer I'll tell my mum that I'm being bullied

5:11 PM
@DavidCarlisle but not on her birthday!

var=$(echo$(wget whatthecommit.com -q -O -) | tr -d '\n' | sed -e 's/.*<p>$$.*$$<\/p>.*/\1/' | awk '{split($0,a,"</p>"); print a[1]}') git commit -e -m "${var}"
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@UlrikeFischer ^^ :)

@PauloCereda Is that cmd or powershell commandline syntax?

@PauloCereda "This should work until december 2013."
Hm....

@PauloCereda -- oh, my. But some teas are absolutely lovely! And coffee doesn't suit Chinese or Japanese food at all! You need to broaden your horizons.

@PhelypeOleinik oopsie :)
@DavidCarlisle Does it speak bash? :)

5:19 PM
@PauloCereda blame @UlrikeFischer if it doesn't

@barbarabeeton @JosephWright was a true gentleman for not beating me up during TUG 2018. I refused tea during breakfast. :)
@DavidCarlisle ooh

@samcarter oh no

@PauloCereda It's much easier to do git commit -em "\$(curl --silent --fail http://whatthecommit.com/index.txt)"

@PauloCereda <3

5:23 PM
@MarcelKrüger ooh <3
@samcarter <3
@DavidCarlisle ^^ see @MarcelKrüger is more efficient :)

@samcarter ooh

@PauloCereda but still bashful

@samcarter the funny thing is that I have co-authored a paper about pedestrian dynamics :)
@DavidCarlisle ooh

Asking a font question for a friend. Are there any script fonts that support the extension of the swash on the capital R? The purpose is to create a logo.

5:27 PM
@PauloCereda ohh! You should write one about the pedestrian dynamics of people running towards cute ducks :)

@samcarter ooh

@samcarter -- This calls into question application of the adjectival meaning of "pedestrian".

@barbarabeeton :P

5:41 PM
@barbarabeeton google suggests a few vv

@barbarabeeton Zapfino or Zapf Chancery?
@barbarabeeton the latter also available as TeX Gyre Chorus =)

@JasperHabicht and @DavidCarlisle -- These fonts do already have obvious swashes, and I believe I've seen multiple lengths of swashes on the Zapfino R. but I can't find a feature that allows extension of the swash. (In other respects, Zapfino is quite attractive.)

5:57 PM
@barbarabeeton so you are looking for fonts that have an opentype swsh feature which affects the R?

@JasperHabicht -- Yes, exactly. Actually, I love the swash on the Zapfino Z (and have a "personal" example of his "HZ" signature, which is just breathtaking). A swash of that sort on the R would work quite well, but getting a font where adjustment of the swash is "automatic" is not easily found, at least not by me.

@barbarabeeton Except it's not an adjective here. :)

@AlanMunn -- True. I'm just being difficult.

@barbarabeeton Join the club.

@AlanMunn -- Oh, I did, long ago. My older sister, were she still alive, would be delighted to confirm that.

6:15 PM
@barbarabeeton it would indeed be great if one was able to search fonts by their opentype features!
@barbarabeeton Agmena Book Italic sold by Linotype seems to have this feature. you can check on their website: linotype.com/de/1110408/agmena-book-italic-product.html
@barbarabeeton EB Garamond also supports this, but it's only italic and not really script. same goes for Agmena Book, though. it seems that script fonts often already have such an swashy R ...

@JasperHabicht -- The examples I'm finding are only "fixed" swashes. Very attractive, many of them, but nothing about extensibility. Thanks for the pointers though.

@barbarabeeton I think, you will only find two or a few alternative glyphs in one font. I don't know about such a feature being implemented as variable font ... but who knows! event this one only has to glyphs: v-fonts.com/fonts/zinzin-vf

@JasperHabicht -- I know that the shape could be created with Metafont or Metapost; it's very like the arabic kashida.

@barbarabeeton that's true.

@JasperHabicht -- Oh, I do love that sassy Q! But the R is relatively pedestrian.

6:29 PM
@barbarabeeton sorry for my multiple posts. i am only getting used to this chat here ... I also found this design.tutsplus.com/tutorials/… (especially the first font that they use as example) but it is still only a lot of alternative glyphs, not totally variable

@JasperHabicht -- No problem. My laptop is misbehaving, and sends off bits before I intend that it should. Nothing too embarrassing yet, but ...

@MarcelKrüger I've been thinking about the tdsdirs stuff: I plan to make an xcopy() function to copy entire directory trees - OK?

@JosephWright Sounds good.

@JasperHabicht -- Now that is definitely worth passing on, and I shall do so. It doesn't provide an exact answer to my question, but it has other useful information. Thanks!

@barbarabeeton you're welcome!

7:14 PM
@JasperHabicht added as feature request for albatross. :)

7:43 PM
@barbarabeeton I know I'm late to the party, but for a logo I would choose a font which overall matches the look and then use a vector editing tool (e.g. inkscape) to adjust the extend of the swash manually.

8:08 PM
@samcarter -- The latest communication I've had is that the individual involved will talk to a retired (reformed?) calligrapher he knows to ask advice. He also suggested that maybe he'll investigate FontForge. He's even more of a fussbudget than I am, so I'm sure that what finally emerges will be attractive.

8:42 PM
@barbarabeeton I just thought that if it is only for a logo, that is, you only need to edit or animate (?) one character, it may be easiest to grab a nice R, get the outlines and then use some vector graphics software to re-shape them? It is even possible to animate vector graphics (SVG) in web browsers with tools such as snapsvg.io

@barbarabeeton you could get someone with natural artistic flair like what I have to design a custom R

@DavidCarlisle Looks like me in the early morning.

Is the layout of the profile changed?

@JasperHabicht -- That's a possibility. I'll pass it along. This person is much more adept at using such tools than I am. My experience has been with function tables, log-log graph paper, french curves, india ink and a ruling pen. Thanks.

@CarLaTeX apparently yes

8:50 PM
@DavidCarlisle More "verbose"

@CarLaTeX I feel it looks a bit strange, because the "Summary" part in the Summary is still quite densely stuffed, but the rest has now a very loose layout ... well, we will get used to it anyways, as we got (probably) used to all the other things ...

@JasperHabicht Yes, of course :) but the previous was more clean

@CarLaTeX lots of space allocated to list all my questions

@DavidCarlisle :D

@DavidCarlisle The answers are now more detailed it seems
@CarLaTeX Now the endless search for that one link to that one function starts again ...

9:01 PM
@JasperHabicht Indeed! :D

@CarLaTeX that's part of the ongoing champaign to make things "responsive"
41

Update: This was rolled out on all network sites December 7, 2021. After receiving and incorporating feedback the community (thank you!) we’ve launched the responsive Activity page! The most common feedback we received was that we’re straying too far from a summary for the summary view. This feed...

@samcarter no comment

@CarLaTeX I think they should sent every user one or more new mice. With so much additional space it now takes much more scrolling to see the same amount of information - this will dramatically increase the wear and tear of the scroll wheel :)

9:17 PM
@samcarter Also the click button is involved, to remove a bookmark now you have to click twice

@CarLaTeX ohh, maybe a secret marketing plot of the computer mouse industry?
4

@samcarter Maybe :D