# Triangulax Interactive Lab

By greenlig

Topic: Maths

[Open the interactive Jig](https://ask.jig.com/j/GXgexncbPcOI)

## Guide

### 1. A lab for bendy surfaces

Soap films, cell skins and rubber sheets are all bendy surfaces\. The triangulax paper teaches a computer to handle them by covering each surface with a net of flat triangles\. On this bench, three helper machines measure the same net so you can compare their answers\.

Source: Silliman, Yu and Claussen, triangulax paper, p\. 1\.

### 2. One shared specimen

Every machine studies the very same specimen on the turntable\. Turn the shape knob, or pick here, to swap between a ball, an egg shape and a ring\. The ring is a torus, a doughnut shape, with the same proportions the paper tested\.

Shapes from the paper's accuracy tests, p\. 15\.

### 3. Finer net, better answer?

A coarse net has big triangles; a fine net has many small ones\. Turn the resolution knob and watch the triangles shrink\. Each step down makes every edge about half as long\.

The paper tests mesh sizes from 0\.25% to 4% of the shape's width, p\. 15\.

### 4. Agent A: the angle rule

Agent A measures how much the surface bends at each corner\. It weighs every neighbour using the two triangle angles that face their shared edge, as drawn on its front plate\. Orange patches on the specimen show where its answer is furthest from the true curve\.

The paper's cotangent Laplacian, equation 3, p\. 5 and Fig\. 3a, p\. 6\.

### 5. Agent B: equal shares

Agent B uses a shortcut: it pretends every triangle is perfect, so every neighbour gets an equal share\. On the ball that works well, but on the egg and the ring the needle swings high\. Making the net finer does not fix it\.

Not from the paper: a comparison helper added for this lab\.

### 6. Agent C: counting corners

Agent C adds up the triangle corners around each point\. On a flat sheet they make a full circle; on a curved surface some angle is missing\. Add up all the missing angles on a ball and you always get one full sphere's worth, however rough the net\.

Angle\-deficit curvature, p\. 5\. Gauss\-Bonnet: the total is exact for any closed net\.

### 7. Compare the machines

Press a machine's top button to make it the active one\. The shared gauge reads its total error against the exact answer, from tiny at the left end to huge at the right\. Try each machine on the ring, from coarse to fine\.

Gauge marks: 0\.01%, 0\.1%, 1%, 10%, 100% error, evenly spaced\.

### 8. Checked against the paper

A green lamp means the machine's results here agree with what the paper reports\. Agent A's error shrinks three to four times each time the net gets finer, and Agent C is exact\. Agent B's amber lamp means the paper makes no claim about it\.

Paper: mean and Gaussian curvature converge in the integrated sense, p\. 14\-15\.

### 9. What the paper builds on top

These curvature tools power everything else in the paper: soft cell membranes, tissue made of tiling cells, waves running round a ring, and rubber sheets that fold into shapes\. Those big simulations are not run on this bench\. They need settings that live in the paper's online notebooks\.

Membrane, vertex\-model and elastic\-sheet demos: pp\. 5\-9\.