Visualizing Functions On Groups

Functions of the form or , where is a group, arise in lots of contexts.

One very natural way this can happen is to have a probability distribution on a group, . The probability density of group elements is a function .

Another way this can happen is if you have some function and has a natural action on ‘s domain – if you care about the values takes at a particular point , you are led to consider functions of the form . For a specific example, the intensity of a particular pixel, , in a square gray-scale image, , subject to flips and rotations, can be considered as a function .

Basic Visualization

Recall that we can visualize finitely generated groups by drawing Cayley Diagrams. (There’s a nice book, Visual Group Theory by Nathan Carter, that teaches a lot of basic group theory from the perspective of Cayley Diagrams.)

The natural way to visualize functions on groups is to picture them as taking values on the nodes of the Cayley Diagrams. One way to do this is by coloring the nodes. In the following visualization of a real-valued function on , dark colors represent a value being close to zero and light colors close to one.

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3D printing of Mathematical Objects!

 

 

 

Back in the fall I did some work on getting hacklab.to‘s 3d-printer to print mathematical objects created in sage. Unfortunately, shortly after I got it to work and printed a test sphere, the 3d printer broke. Thus began a long succession of the makerbot — nicknamed the break-r-bot — being fixed and broken… spending most of its time broken.

But recently it was fixed and I decided to dig out my old code and get to work on it.

Separation Axiom Visualisations

A couple days a go, I saw some nice visualisations of separation axioms on Wikipedia. Unfortunately, it wasn’t a full set. Well, here is a full set (well, T0, T1, T2, T2 1/2, T3, T4, T5):

Compactness Graph

Here’s the first revision of a graph of the implications of topological properties that I made:

(Click on it to see a better version!)

It’s mostly based off the stuff in Counterexamples In Topology (great book, BTW) but I did add some stuff (like Baire!) and merged/reorganised it. Diagram was made by Graphviz.

Most of the implications are trivial, but there are a few I haven’t prooved yet (most of the ones involving seperation axioms).