isomorphs

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Pyrargyrite, Ag3SbS3 (top) and proustite, Ag3AsS3 (bottom).

These are isomorphous minerals, meaning the atoms have the same arrangement in the two compounds, but where pyrargyrite has antimony, proustite has arsenic.

Antimony and arsenic are in the same column of the periodic table, so we would correctly expect them to make this swap without much fuss.

This little gem is dedicated to Frédéric Vanhove, who passed away yesterday and was our assistent for graph theory. He had a passion for graphs, so I hope he’d like this beautiful theorem - in my opinion one of the most elegant in whole mathematics.

Draw n points and connect them without creating any “loops”; in the result, every point should be accessible from every other point by exactly one path. Such a configuration (a graph) is called a tree. You can see all possible trees on four points in the image. Lots of these trees are essentially the same (isomorphic), but we label the points to distinguish between them.

Carl Wilhelm Borchardt found an elegant formula to express the total number of possible spanning trees on n labeled points, but nowadays the result is named after Arthur Cayley. In our example (with n=4) we find 16 trees, but in general, the formula states that this number is exactly nn-2.

Shortly, nn-2 is the number of spanning trees of a complete graph Kn.

Ammonites are a group of extinct cephalopod mollusks with ribbed spiral shells. They are exceptionally diverse and well known to fossil lovers. Researchers have developed the first biomechanical model explaining how these shells form and why they are so diverse. Their approach provides new paths for interpreting the evolution of ammonites and nautili, their smooth-shelled distant “cousins” that still populate the Indian and Pacific oceans.

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"Draw all non-isomorphic, homeo- morphically irreducible trees on 10 vertices. Explain why all such trees are represented among your drawings."

(Editor’s note: The answer to the math question in Good Will Hunting looks a lot like an Eames pattern.)

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i was tagged by the beautiful silllystyles to post six selfies. i’d say the common theme here is lipstick and an overuse of filters.

i’m a bit late to the party, and am not sure who’s done this yet, but i’ll (non bindingly) tag isomorphic, foxsmoulders, durancerbecomewords, maggielawson, sofoxsea, & selfishandshrewd!

A mathematician has developed a new way to uncover simple patterns that might underlie apparently complex systems, such as clouds, cracks in materials or the movement of the stockmarket. The method, named fractal Fourier analysis, is based on new branch of mathematics called fractal geometry. The method could help scientists better understand the complicated signals that the body gives out, such as nerve impulses or brain waves.

Group Theory, Isomorphisms & Permutations

I’m not feeling social interaction today so I learned a bunch of group theory, and it just got too cool, so i had to post some of it on here!

Now, before I go on to talk about the main things which are Isomorphisms and Permutations, I should properly define groups.

Taking the axiomatic approach, groups are sets with an operation for combining elements which pass the following tests

Groups are closed under one operation, meaning if you combine any two elements using an operation of your choice (the symbols +and * are often used), you will never be able to create an element which is not in the group.

The operation is associative meaning A*(B*C)=(A*B)*C=A*B*C as long as you combine the elements as shown. This effect can be extended to any string of n elements. 

There exists an element “e” such that x*e=x and e*x=x. This element is called the unit, and if we are talking about multiplication with ordinary numbers, this number would correspond to one.

For each number in a group, there will be an inverse. When we are using multiplication-like operations like we are above, we usually write the number like this a-1, and call it “a inverse” for addition we use -a and call it “the negative of a”. The inverse of the unit is the unit again, and any element combined with its inverse becomes the unit.

You can see that the commutative axiom is not listed, where A*B=B*A this is not always true, as we will see in a second. Take the group consisting of the elements

 

 


 


under the operation of matrix multiplication which works like this.

  Now, if we do the operation A*B, we get the matrix C, if we do the operation B*A, we get the matrix K, meaning that this operation is not commutative. This is not the only operation which behaves this way, so in general we have to say that the operation does not have to commute. If it does, we call the group an Abelian group.

Now that that’s out of the way, we can define an Isomorphism. 

An Isomorphism is a one-to-one function which takes one group and converts it into another while preserving the operation. Basically, if G  is a group under the operation * denoted <G,*> and <H,+> is a group, and 


then an Isomorphism f(x) will take f(a)=a’ and f(b)=b’ and make, f(a*b)=a’+b’ or, entirely in function notation, f(a*b)=f(a)+f(b).

To prove that two groups are isomorphic, you need to prove that the function you are using is one-to-one, that is proving that the function is both injective and surjective. To do this, you have to prove that each element of the range is the image of at most 1 element of the domain, which can be done by showing that it fits this definition 

 
You then have to prove that the function is surjective by showing that for some element y in he range, there exists an element who’s image is y. Together, these two things prove that the function is bijective, which means that you can turn elements from G into elements in H and vice versa.

From there, all you have to prove is that f(a*b)=f(a)+f(b) and you’re done.

Now why are isomorphisms important? Because they can be used to prove things like this Cayley’s theorem which says that every group is isomorphic to a group of permutations.  Quite a bit is known about permutations, which makes this theorem very strong in terms of its use.

We can prove this in a second, but first, we have to look at permutations. 

A permutation is a function on a set S which takes the elements of S and rearranges them. Now, this must by definition be a bijective function, which takes S and maps it to its self. Taking individual permutations to be elements of a set, we can create a set of n! elements, where n is the number of numbers being used in the permutation. For instance, the set of all permutations on three numbers is:

  

 

 

 

and has 3!=6 elements. To combine the elements, we define the permutation of a permutation to be a function which takes the elements as they are arranged by the inner permutation and then rearranges them under the outer permutation called the composition. For instance:


To show that this combining of permutations is closed under the operation above, we can make a chart like the one below, where the first column is multiplied on the left by the first row,

but there are an infinite number of groups of permutations, so we must prove it. We can actually take it a step further into abstraction, and prove the composite of any two functions which are both bijective will yield another bijective function. Why are we proving the combinations of permutations are bijective? Because we started with the assumption that we had made a list of all the possible permutations for n objects, so showing that this new function is also bijective permutation means that it was on this list in the first place. 

Assume a f(x) and g(x) are injective functions, then we have to prove that [f*g][x]=[f*g][y] implies that x=y. Well suppose  [f*g][x]=[f*g][y], then f(g(x))=f(g(y)) because f(x) is injective, g(x)=g(y), and because g(x) is injective, x=y, so [f*g][x] is injective

Assume g:A ->B and f:B ->C are surjective, then we have to prove that every element of C is f*g of some element in a. Assume z is an element of C, then because f is surjective,  f(y)=z for some y in B. Because g is surjective,f(g(x))=z so, for any element of C, there is an element for which z is its image under f*g, so [f*g][x] is surjective.

Now, assuming the functions f and g are both bijective, it follows from what was just proven that f*g is bijective as well.

By this, the group of permutations under composition is closed.

There exist inverses for each element, which return each permutation to the permutation


which also happens to be identity element. Finally, combining permutations is associative, so we have a group.

Now we can prove Cayley’s theorem which again states that each group is isomorphic to a group of permutations.  

Proof

Begin by assuming we have a group G, we wish to prove that it is isomorphic to a set of permutations. Well, permutations require a set, so why not use the set that G is built off of? The only set we have around is the one used to make G, so we have no other choice but to use it. We then define a function


which is defined as:


And gives all the permutations of G, changing each element a of G into ax, and then ranging X over all elements of G to give one permutation per element. 

This function is injective, that is 


and surjective, as for some y in g, 

 
so y is the image of some element a^(-1)*y.

So we have a bijective function which turns an element a of G into a permutation which turns x_1 into a*x_1, x_2 into a*x_2 and so on. So we let G* represent the set of permutations of G created by pi_a(x) as a ranges through the elements of G. 

Notice now that the set of permutations created is not necessarily the set of all permutations of G, but a permutation corresponding to each element. We now have to prove that it is a group, that is


and that for any permutation contained by G*

 
Where pi_e is the identity permutation given by multiplying each x in G by the identity. The associative property can be assumed due to the fact that the operation is simple non-commutative multiplication. 

To begin, we have to prove that 


We can say that 


Because ab is a member of G, we can say that G* is closed with respect to composition. 

Finally, because G is a group with inverses, 


So we have proved that G* is in fact a group.

Now all we have to do is prove that G is isomorphic to G*. We already have the function ready, so we use it as the isomorphism. 


It is injective,


and surjective, every element pi_a of G* is the image of some f(a)

Lastly, 


Thus, f is an isomorphism, and

That’s about all I have left in me for today, but I’ll try to get more up on here tomorrow! I thought Cayley’s theorem would be a good start!

#221

Bad conlanging drinking game:
Find a work of fantasy or sci-fi, perhaps a book or a fan-fic or the like

-Shot if unnecessary use of apostrophe

-take a swig if realize morphemes are isomorphic to English

-shot if use word in conlang that means exact same thing as a word in English, for no apparent reason

-two shots if talks about pun that works because of homophones in English

-shot if language is inconsistent (same name spelled multiple ways for no reason, e.g.)

-shot if uses quirks of Latin orthography for no reason (<c> versus <k> versus <q>)

-swig if “Original script” is in oddly one-to-one correspondence with Latin alphabet

-sip for each diacritic that means nothing

-swig if whole bits are stolen from other languages

-three shots if author uses “guttural” to mean “evil”

-a shot for each instance of “they have no word for [concept], so cannot think of it,” “these people have words for [concept] that allow them to see the world better than other people,” or other Sapir-Whorfery

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I just have lots of Tron feels… xD I did this comic a while back when I found out what a NAVI Bit is. The concept of ISOs being undirected and lost got to me. I think Quorra’s NAVI would’ve gotten very cross with her for always sneaking off into places where she shouldn’t go—which would’ve been funny to make into a comic, buuut this happened instead. Cuz feels. xD Enjoy! <3

isomorphismes of 2014

Well, almost all the planned isomorphismes of 2014 are still in drafts. Here’s what I did write this year:

And some posts worth re-advertising:

Besides Contravariance, covariance, and Reading History Sideways and other drafts I mentioned last year, I’m trying to write something about the creation of desire. And what it means that dating sites ask your favourite films. And how much is the parameter space of career choices like an n-sphere?

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i was tagged by oliverjonasqveen for the 2014 in selfies challenge! idk how many i was supposed to post so i may have gone a little overboard but w/e w/e i take a lot of selfies we know this! as always go ahead and click ‘em for some captions so you can mock me for it later woohoo!

i’ll go ahead and tag: juliagrahams isomorphic bluestoplights amerikates jamesnowday captainduckling

for more of my face: 2013 in selfies (same dumb face, different year)

The function of [Hitchens’] antitheism was structurally analogous to what Irving Howe characterized as Stalinophobia…the Bogey-Scapegoat of Stalinism justified a new alliance with the right, obliviousness towards the permanent injustices of capitalist society, and a tolerance for repressive practices conducted in the name of the ‘Free World’. In roughly isomorphic fashion Hitchens’ preoccupation with religion…authorized not just a blind eye to the injustices of capitalism and empire but a vigorous advocacy of the same.
—  Richard Seymour, as quoted in Jacobin