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:

To begin this story we need to think about an equilateral triangle:

This is a bit of a cheat for a triangle as the edges are curved, but the edges do form an ‘S’ starting and stopping in the same direction, so as we go round the triangle the only place we turn is the three corners. Each of these has six symmetric arms. So the angles will be 60 degrees. Three such angles of course gives $180$. Extending this triangle out gives a pretty, flat plane pattern:

What happens if we take an arm away? Just five arms at each corner, the triangle now has $3 \times 72$ degrees, which is $216$ – a little too much:

As you can see this bad triangle now does not lie flat but bends up. Now when we keep connecting triangles the result continues to curve round

Building up into a sphere:

This will be familiar to many as a spherical icosahedron. Unlike the polyhedral version, with flat faces and sharp edges and corners, this one bends smoothly into a sphere. The curvature of the surface happening in the gaps between pieces. Now we can look at that extra angle, each triangle had an excess of $36$ degrees. There are 20 triangles so the total spare angle is $720$ degrees. Keep that number in mind.

Let’s take another branch away:

With four arms the angle is 90 degrees. So the triangle now has $3 \times 90= 270$ degrees, an excess of $90$. Building out with these triangles we again get a ball:

This new ball has eight triangles, so the total excess is $8 \times 90$, again $720$. This is not a coincidence, and is related to one of the earliest results from what would become the study of topology, Descartes’ theorem. If we started with 3-branch pieces and again made triangles can you work out how many triangles we would end up with?

You can play the same game with a torus like this one, named in a bad pun Taurus, one of the kits you can buy. (We did say that the system would make a good wreath!):

This gets a little more complicated as it has triangles and quadrilaterals, however the final sum adds up not to $720$, but to $0$. That is because some of the shapes have too little angle, not too much and everything balances out. What happens when every shape has too little angle? A simple way to do this is to extend the sequence $4,5,6$ by $1$, starting with 7-branch pieces:

The result again curves, but not in a single direction like the previous examples, it waves up and down, not unlike a saddle. Continuing with more triangles the surface gets more wobbly:

This is not just a saddle, but a surface with a saddle at every point, so as you build out the amount of material added gets greater and greater, creating the surface we started with:

This object is a model for the mysterious mathematics of hyperbolic geometry, as well as giving hints for how nature makes surfaces with a high surface area to volume ratio like Corals or our brains. Yet we can start to think about it just by connecting some triangle pieces together so that everything lies flat locally.

What will you explore with Curvahedra?

]]>More information at Wired.

*via math-fun.*

Apparently those symbols winding their way around the garden are “plant growth algorithms”, whatever those are.

There’s also a golden-ratio-thingy water feature, of course.

You can thank Winton Capital, sponsors of all sorts of worthy maths projects, for this bit of mathsy art.

]]>Marcus du Sautoy has teamed up with animator Simon Russell to create this animatino to accompany Messiaen’s *Quartet for the end of time*. It’s got all the usual arty maths things in it – the Fibonacci sequence and golden ratio, prime numbers, polygons and polyhedra of all sorts – as well as the less well-trodden sporadic group $M_{12}$. It all comes together quite nicely, though I much prefer the elegant end to the spiky-frenetic start.

There’s a page describing all the maths ideas to be found in the video at Sinfini Music.

*via Marcus du Sautoy and Sinfini Music on Twitter*

*via Colossal*

Chris Watson has written in to tell us about his site, *Tessellation Art*, where he sells his heavily Escher-inspired prints. They’re available in a range of sizes and media, and quite affordably priced. I particularly like the print above, titled *Vortex*.

This is a really nice idea. *Le Livre de l’Incomplétude* (*The Book of Incompleteness*) is an “artistic appropriation of Gödel’s incompleteness theorem,” initiated by artist Débora Bertol. The superficial understanding of that theorem is that every consistent formal theory contains truths which can’t be proved inside that theory, so the book’s conceit is that it will catalogue as many different arithmetic formulas as possible that evaluate to each of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

I think it’s a really charming take on one of the most abstract and hard-to-understand subjects in maths.

When you go to the site, you’re presented with a column of forms, one for each number. You can submit as many formulas as you like for each number; clicking the **?** button submits what you’ve entered for verification. Valid formulas are immediately tweeted by @l_incompletude, and once there are at least 100 formulas for each number they’re going to start publishing an ebook.

The claim that each proposed solution is verified automatically made my pedantry finger twitch, so I of course immediately entered the following:

Gödel’s first incompleteness theorem states that a system is either incomplete or inconsistent; the Book of Incompleteness seems to have plumped for inconsistency.

… which sort of undermines the project. So maybe it’s just a bug.

Anyway, thinking up new ways of expressing the same number is a relaxing, meditative exercise, like raking a zen garden. I might spend a while submitting $0 = n-n$ for different $n$ for a while.

@l_incompletude on Twitter

]]>John Edmark has 3d-printed a series of sculptures which do something rather remarkable when you rotate them. In the stop-motion animation above, the sculpture rotates by the golden angle in each frame.

**See more:** Blooming Zoetrope Sculptures by John Edmark at Instructables.

*via Henry Segerman on Google+*

It’s a Flash applet, which means it doesn’t work on mobile devices :(

**Play **Doodal

Erik Åberg is selling a short documentary about these lovely foldy cubes on his website.

*via Will Davies on Twitter*