## Tetrahedral symmetry

I was curious how one would go about rendering a triangle group using Möbius transforms. It took a while! I wanted to jot down the underlying theory so I wouldn’t forget it later. Perhaps these notes will be of interest to others as well. Here goes: Given a triangle with angles π/l, π/m, π/n, you…

## Congruence Group Γ(2)

Modular Group Γ(1) generated the hyperbolic tessellations seen previously. Its generators are: Or in matrix form:  [0,1;-1,0], [1,1;1,0]. The Congruence Group Γ(2) can be generated by these matrices: [1,2;0,1], [1,0;-2,1] The tiling looks like this: You can view a version I wrote in Three.js/WebGl here. You can spin it and pan and zoom. It is clear…

## Ford Circles and Farey Graphs

Here is an image of a tessellation or tiling of the upper half-plane. Under some group of symmetries, an initial triangle (for instance the one in orange) covers the plane without overlaps. (The initial triangle is not ideal because only one of its angles is 0). The group of symmetries used are those of the…

## Tessellation of the Hyperbolic Plane on the Riemann Sphere

I’m interested in learning more about the modular group and this article looking into hyperbolic tessellations represents my initial efforts to collect my thoughts on the topic. We are familiar with hyperbolic tessellations from the artwork of M.C. Escher and from numerous renderings of the Poincare Disk (which was really invented by Beltrami, not Poincaré):…

## Crib Sheet on Mathematical Groups

This is just a bunch of quotes I’ve pulled off of Wikipedia and elsewhere. I tried to get a handle on some of the basic groups used in hyperbolic tilings and complex analysis. ——————– In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the…

## Möbius Transformations on Spherical Photos and Videos

Spherical Video presents some interesting challenges, for instance, how do you zoom? Or how do you rotate on any arbitrary axis? As the mathematician Henry Segerman pointed out in a post for EleVR, you can achieve both of the above using Möbius transformations. The transformations are conformal (they preserve angles), and they map circles to…

## Thought Bubbles

A window is a two-dimensional hole in a two-dimensional plane that allows you to see into a three-dimensional world. So what if we could make three-dimensional holes? The original idea came from a three.js demo by altered qualia where he was demo’ing fresnel shaders: I didn’t care much about fresnel shaders, but was intrigued by…

## Complex Surfaces

I wrote a separate version of formula toy that handles complex functions. So, you can type in a function like: f=sqrt(g). Both f and g are expected to be complex functions:     g = u + iv and     f = w + ix where u is the real component of g, v is the…

## Torus Knots

Formula Toy is a simple and free WebGL app I wrote that allows you to enter in 3-d formulas and see the resulting surface. Sort of like Desmos, but for 3D. When I first wrote it you could express your formulas in 3 different coordinate systems: cartesian, spherical (polar), and cylindrical. I recently add toroidal…

## Rotations, Transformations – Geometries and Meshes

I was driving myself batty trying to get all my rotations and transformations to behave correctly in three.js. So in these kind of cases one must always pare down to essentials. As follows: Lets create a simple mesh and place it on the scene: // Example 1 var geo = new THREE.BoxGeometry(5,5,20,32); _mesh = new…

## Drawing Pentatope Cross-Sections in three.js

A triangle is the simplest regular figure in 2 dimensions. Its 3 dimensional analogue is a tetrahedron. Its 4 dimensional analogue is the pentatope. Another term for these 3 geometries is: simplex. Many articles online explain these further. I wanted to view what it would look like if a pentatope passed through our 3-dimensional space.…

## Minecraft Menger Sponge – STEAM project.

What has zero volume and an infinite surface area? A Menger Sponge of course. If your child is a Minecraft fan like my 7 year old, then this simple fractal can be used to make a good afternoon STEAM project that teaches math and programming concepts. (Note you should already know how to modify Minecraft…

## CVision: Computer Vision Workbench

CVision: A computer vision WorkBench and Utilities written in WinForms that wraps OpenCV. I wrote it for my purposes, there is plenty to improve: https://github.com/rwoodley/CVision. Features: Has all OpenCV color options and color maps. Histogram Equalization 4 Blurs Many Morph Modes: ERODE, DILATE, OPEN, CLOSE, GRADIENT, TOPHAT, BLACKHAT You can specify morph type (RECT, CROSS,…

## The Visual Representation of High Dimension Spaces

Our brains struggle visualizing spaces with more than 3 dimensions. This is a problem we tried to address in our FaceCloud and FaceField projects. We present here a possible solution using fractional dimensions to represent higher dimensions. The examples here are all based on the Eigenfaces face recognition algorithm where we were dealing with high…

## Face Field Update

10 Synthetic Faces are now on display at the Chicago Cultural Center through August, as part of Adelheid Mers’ “Enter the Matrix” exhibit. They look great in large format and are quite powerful. Some photos below. These are faces pulled at random from the 60-dimensional PCA space that we have been working with for some…

## Formula Toy

Back at the age of 17, thanks to the liberal access policies of Indiana University’s Wrubel Computing Center, I was able to write short little computer programs (on cards!) that would graph 3 dimensional surfaces using a pen plotter. The little nerd that was me was thrilled at the results, and my bedroom wall was…

## The Hairy Blob, 800 Ping Pong Balls, and a Mindstorms RoboCam

The Hairy Blob: At the Hairy Blob exhibition at the Hyde Park Art Center last spring, visitors were invited to draw an image of time on a ping pong ball and toss it into a net that was suspended from the ceiling.      800 Ping Pong Balls: What to do with over 800 ping…

## Anti-Face Model Specification and Calculation Details

Update 10/2013: We have implemented this in a free iPhone app which is available here: https://itunes.apple.com/us/app/anti-face/id690376775   Our site FaceField.org (and now the iPhone app) uses the EigenFaces methodology as implemented in Open CV to calculate a special kind of face that we have labelled an ‘Anti Face’. Model specification: – Over 1000 faces were…

## Visualizing Factors and Prime Factorization.

Update 8/2014: The calculator discussed below is also available in the Chrome store here (for free of course). I’ve previously discussed Brent Yorgey’s factor diagrams. As the father of a 6 year old, I’ve found they are a great way to introduce the concepts of primes and factorization. Since then, I dabbled with the javascript…

## Factor Dominoes

Many have discovered Brent Yorgey’s very cool factorization diagrams. They seem like a great way to teach multiplication and factorization to children. We also were excited by the domino game suggested by Malke Rosenfeld on her blog and decided to try to take her ideas and see if we could create viable game that also…

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