## You're reading: Posts Tagged: hyperbolic geometry

### Curvahedra Geometry

Longtime friend of the Aperiodical, artist, mathematician and #BigMathOff semifinalist Edmund Harriss has come up with a new puzzle/toy/exploration set, developing his Curvahedra system. We asked him to explain the maths behind it in this guest post.

Curvahedra is a flexible system of connectors that can make all sorts of different things, combining puzzles (and self-created puzzles) with art. You can get your own to play with, explore, prepare for Christmas (they make great decorations, wreaths and presents) at our online store, and get 15% off with the discount code APERIODICAL.

As this is the Aperiodical, you might be most interested in how it can be used to explore mathematics. In the big math off I talked about the basic ideas behind the system, Gauss’ famous Theorema Egregium and Gauss-Bonnet theorems. A really simple version of this comes from just considering triangles, that can be built up to make this:

### 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.

### Aperiodical Round Up 6 – It glides to a stop as it reaches the end of the power stroke

Hello. I’m Christian Perfect and it’s finally here: Aperiodical Round Up 6!

It’s certainly been a while since the last Round Up. You might not even have the words to describe just how long it’s been. Maybe the book Naming Infinity will help.

### Newcastle MathsJam April 2012 Recap

April’s MathsJam was very enjoyable. We did a bit of arts and crafts, a bit of playing games, and if it had been NBA Jam instead of Maths Jam I would have been entirely on fire because I used up all my IQ points solving some very fun puzzles. Durham were still on their Easter holidays so the attendance was a modest six people. That was just enough for everyone to be doing the same thing at the same time, so we had a good time.

Continue reading “Newcastle MathsJam April 2012 Recap” on cp’s mathem-o-blog

### HyperRogue II – a roguelike on the hyperbolic plane

I was directed to this game by a retweet by @haggismaths. It’s a roguelike (text-based explorey role-playing adventure game) which takes place on the hyperbolic plane. It’s a lot of fun. It’s hard to get your head round the fact that there’s a lot more stuff in between two lines in hyperbolic space than in Euclidean space, so it’s very hard to find your way back somewhere after it disappears over the horizon.

You can download a windows executable, or source code which will compile on Linux, at http://www.roguetemple.com/z/hyper.php

### Conformal Models of Hyperbolic Geometry by Vladimir Bulatov

Vladimir Bulatov makes art, including metal sculptures and jewellery, based on tilings of non-Euclidean spaces.

He has posted online some slides he made to go with a talk he gave at the JMM in 2010, about the many ways conformal mappings of the hyperbolic plane can produce interesting images. Quite a few of the diagrams are animated if you click on them, which I missed first time round.

There are a few other slideshows on similar topics on his site.