This is the **158th** **Carnival of Mathematics**, a monthly round-up of interesting maths bits from across the internet. Convention dictates that I now therefore specify some interesting facts about the number 158. Unfortunately I am writing this on a train with no internet access, which will make fulfilling this obligation more than usually challenging.

# You're reading: Posts Tagged: polyhedra

### Curvahedra is a construction system for arty mathsy structures

Edmund Harriss is a very good friend of the Aperiodical, and a mathematical artist of quite some renown. His latest project is CURVAHEDRA, a system of bendable boomerang-like pieces which join together to make all sorts of geometrical structures.

### Maths Object: Nobbly Wobbly

My maths object this time is one of my dog’s favourite toys: the Nobbly Wobbly.

In the video, I said it was invented by a mathematician, but Dick Esterle’s bio normally goes “artist, architect, inventor”. I’ll leave it up to you to decide if Everyone’s a Mathematician.

It’s a particularly pleasing rubbery ball thing made of six interwoven loops in different colours, invented by Dick Esterle.

On Google+, various people told me the unexpected fact that the outer automorphism group of $S_6$ is hiding inside this dog toy.

I’ve also found this Celebration of Mind livestream starring Dick Esterle from 2013 talking about all sorts of mathematically-shaped toys, including the Nobbly Wobbly.

### Maths Objects: polyhedra

Time for some more maths objects! This time I wanted to show you the various polyhedra I’ve got around my desk.

- The tetrahedron is made out of a paper plate, following the instructions on the brilliantly kooky wholemovement.
- The sonobe cube is a classic. Mine’s made out of Post-It notes.
- The swirly thing is made out of curler units. Here’s a nice lady explaining how to use them to make a few different polyhedra.
- The classic reference for the Post-It note dodecehadron is James Grime’s video instructions.
- Once you can make a dodecahedron, add some more maths by edge-colouring it. I followed Julia Collins’ 5-colouring. Or if you’re more adventurous and less colourblind, look at George Hart’s colourings page for some really sophisticated patterns.
- I can’t remember how I made the icosahedron. Can anyone remind me?
- Finally, I’ve shown off the enneahedron loads of times. I wrote about its creation a couple of years ago.

### Review: Unique polyhedral dice from Maths Gear

Our good friends at Maths Gear have sent us a tube of “unique polyhedral dice” to review. The description on mathsgear.co.uk says they’re “made from polyhedra you don’t normally see in the dice world”. My first thought was that we should test they’re fair by getting David to throw them a few thousand times but — while David was up for it — I’d have to keep score, which didn’t sound fun.

So instead we thought of some criteria we can judge the dice on, and sat down with a teeny tiny video camera. Here’s our review:

### Have fun playing with curvature

Recently Tim Hutton and Adam Goucher have been playing around with hyperbolic tesselations. That has produced a {4,3,5} honeycomb grid for the reaction-diffusion simulator *Ready*, which Adam talked about on his blog a couple of days ago. Tim has also made a much simpler toy to play with in your browser: a visualisation of mirror tilings (the Wythoff construction) in spaces with different curvatures.

*Hyperplay* lets you select the kind of regular polygon you want to tile, and then your mouse controls the curvature of the space it sits in. Certain curvatures produce exact tilings of the space – for example, triangles tile a space with zero curvature – and you get projections of polyhedra for certain positive curvatures.

### Radii of polyhedra

*(At last month’s big MathsJam conference, we asked a few people who gave particularly interesting talks if they’d like to write something for the site. A surprising number said yes. First to arrive in the submissions pile was this piece by Tom Button.)*

The formula for the surface area of a sphere, $A=4\pi r^{2}$, is the derivative of the formula for the volume of a sphere: $V=\frac{4}{3}\pi r^{3}$.

This result does not hold for a cube with side length $a$ if the surface area and volume are written in terms of $a$. However, if the surface area and volume are written in terms of half the side length, $r=\frac{a}{2}$, you get the surface area $A=24 r^{2}$, which is the derivative of the volume, $V=8 r^{3}$.