The Butterfly (Foldables 1)

We all know that cardboard cubes are rigid, which is why we get our packages in boxes. We also all know that if we remove two opposite faces from a cube, we can fold it together. This started to interest me when I noticed that the polyhedral approximation of the Schwarz P surface is surprisingly flexible. This summer, I showed this to our local Origami and Paper Folding expert, Jiangmei Wu from our School of Art and Design, and she became interested. A few days later she came with a paper model that looked like this:


She called it a simple variation of the polyhedral P-surface. Hmm. This is a triply periodic polyhedral surfaces tiled with rhombi. To understand it, we build it out of smaller units (which we called butterflies):


The really cool thing about it is that it can be folded together in two different ways, like so:


You can find an animation showing the continuous deformation here. We stared at this for a (long) while, until we realized that this has to do with rhombic dodecahedra. The structures up above are composed of the rhomboids from last week that tile a rhombic dodecahedron. The latter has, as the name hints, 12 faces, which occur in opposite pairs. Like the cube, it is rigid per se, but becomes foldable if we remove two pairs of parallel faces, leaving us with four faces to use, which are distinguished by color up above.Fractal0b

Above you can see the four hollow parallelepipeds (which we called hollowpeds). The almost trivial but nevertheless mind bending realization is that everything you build out of these hollowpeds becomes a structure foldable in two different ways. Next week I’ll show Jiangmei’s second model, a foldable fractal… If you can’t wait, check out this.




Beach Balls

Following Jack’s suggestion, let’s try to visualize holomorphic maps from the Riemann sphere to itself. As a reference, here is the Riemann sphere with polar coordinates, and on the right hand side its cylindrical projection.


The preimage of this under a quadratic polynomial looks like this:


And here a rational function function of degree 4:


One can get quick images using uv-mapping, but there are some rendering artifacts I don’t know how to get rid of yet. Degree 5:


Finally, the Gamma function, which has an essential singularity at infinity.


Martha says their gypsum printer can print these in color. We’ll see, I hope.

Julianna and Friends

I have written before about Sofia, one of the wonderful cheeses from Capriole Farm.

DSC 1220

Like her, Julianna (up above) is made from goat cheese, but comes with a nice herbal crust. It’s the stronger companion of the Old Kentucky Tome, which you find below to the right.

DSC 1218

There are other things from here I would like to take with me to my next life, whenever this will happen. The bread, for instance. American bread used to be the biggest nightmare in this country. Not anymore. One reason is the Muddy Fork Bakery that produces this Rustic Sourdough with a perfect crust,

DSC 1216

or the beautiful Sesame Spelt

DSC 1217

that goes so well with the goat cheese. All their breads are hand-made and wood fired. Amazing stuff. You can find both bread and cheese at the local Farmer’s Market or at Bloomingfoods.

DSC 1215

Stellated Triacontahedron

If you have mastered the Slidables from last year and had enough of the past gloomy posts, you are ready for this one.

Let’s begin with the rhombic triacontahedron, a zonohedron with 30 golden rhombi as faces. There are two types of vertices, 12 with valency 5, and 20 with valency 3. In the image below, the faces are colored with five colors, one of which is transparent.


The coloring is made a bit more explicit in the map of this polyhedron below.

Triagraph 01

We are going to make a paper model of one of the 358,833,072 stellations of it. This number comes from George Hart’s highly inspiring Virtual Polyhedra.


In a stellation, one replaces each face of the original polyhedron by another polygon in the same plane, making sure that the result is still a polyhedron, possibly with self intersections.

Newface 01

In our case, each golden (or rather, gray) rhombus becomes a non convex 8-gon. The picture above serves as a template. You will need 30 of them, cut along the dark black edges. The slits will allow you to assemble the stellation without glue. Print 6 of each of the five colors:

DSC 8159

Now assemble five of them, one of each color, around a vertex. Note that there are different ways to put two together, make sure that the original golden rhombi always have acute vertices meeting acute vertices. This produces the first layer.

DSC 8161

The next layer of five templates takes care of the 3-valent vertices of the first layer. Here the coloring starts to play a role.

DSC 8162

The third layer is the trickiest, because you have to add 10 templates, making vertices of valency 5 again. The next image shows how to pick the colors to maintain consistency.

DSC 8166

Below is the inside of the completed third layer.

DSC 8168

Two more to go. Layer 4 is easy:

DSC 8170

The last layer is again a bit tricky again, but just because it gets tight. Here is my finished model. It is quite stable.

DSC 8174

Another Brick in the Wall

When Apple announced in July this year they had sold 1 billion iPhones, I started wondering about another brick maker: How many blocks has Lego made? Their friendly customer service couldn’t tell me how many elements they have made in total, but the yearly production is 19 billion. Scary. Unfortunately, the shape of the standard lego brick is too limited for my needs. For a long time, I had wanted a lego brick in the shape of a rhombic dodecahedron (better would be a four dimensional lego hypercube of which the rhombic dodecahedron is a mere shadow, but let’s not be delusional). As you can see, this polyhedron tiles space as well if not better than the cube.


Various companies have produced shapes with more or less cleverly embedded magnets, but keeping track of the polarity on all faces of a 12 sided object is tricky. And this would be a lot of magnets. The actual problem, however, is the enormous amount of choices one has: 12 faces to attach to is just too much. I strongly believe that Lego’s success stems from the fact that they have reduced the number of possible ways how you can attach two lego pieces dramatically. No choice means dictatorship, two choices US capitalism, but more choices sounds like European liberalism or even anarchism, and we see where that leads.

This gave me the idea to replace the complicated rhombic dodecahedron by a simple object that is less attachable. Here is the new brick.


To make it, take three faces of the rhombic dodecahedron that are symmetrically positioned, and replace each of the three rhombi by its inscribed ellipse. Then take the convex hull of the ellipses. The resulting shape consists of the ellipses, two equilateral triangles in parallel planes, and three intrinsically flat mantel pieces.

You will notice that there are two versions of this brick, a left and a right handed one. This leaves just the right amount of choices.


If you alternatingly attach a left to a right brick, you get a hexagonal annulus. Remember that we are still tiling space using slimmed down versions of the rhombic dodecahedron. Due to our imposed limitation of choice, nor every place can be reached anymore. The hexagonal annulus is a little simplistic. What do we get if we just use the left handed brick?


Let’s start with a red central brick, attach a brick on all three sides, and another six at the free faces of the new bricks. We notice that the bricks can occur in four different rotated positions. I have distinguished them by color. Add another 12 bricks:


And another 24. No worry, no intersections can occur, because, I insist, we just tile a portion of space with rhombic dodecahedra.


Now we see that the tree like structure we have produced so far does not persist. In the next generation, we obtain closed cycles of length 10, and we finally recognize the Laves graph.


In the very near future you will see what else one can make with these bricks.

Ragged Rectangles (From the Pillowbook II)

In a ragged rectangle, the sides zigzag diagonally as in the left figure below, which shows a ragged rectangle of dimensions 6⨉7, and within a ragged 3⨉3 square. Note that the boundary changes directions at every unit step. These shapes make interesting candidates for regions to be tiled with polyominoes. The example in this post illustrates nicely how the interplay between making examples and generalization leads to a miniature theory.

Raggedex 01

To tile a shape like this with polyominoes, it will help to know its area in terms of unit squares. This is easy: If you color the squares in a ragged a⨉b rectangle beige and brown, you will get a⨉b squares of one color, and (a-1)⨉(b-1) squares of the other color.

This right away shows that it is hopeless to tile a ragged rectangle with dominoes. The first really interesting case is to use L-trominoes. The area formula implies that we need one dimension of the rectangle to be divisible by 3, and the other to leave remainder 1 after division by 3. Thus the shortest edge that can occur has length 3, and the other them must have length 3n+1. The figure below shows how to tile any ragged rectangle of dimensions 3x(3n+1) with L-trominoes:


The next shortest edge possible has length 4, and then the other edge must have length 3n. Again, a few experiments lead to a general pattern which shows that any 4x(3n) ragged rectangle can be tiled with L-trominoes:


This covers the two basic kinds of thin and arbitrarily long rectangles. What about larger dimensions? If we already have a ragged rectangle tiled with L-trominoes, we can put a frame around it that is also tiled with L-trominoes:


These three constructions together show that a ragged rectangle can be tiled with L-trominoes if and only if its area is divisible by 3. Next time we will see how this helps us to tile curvy rectangles with pillows.

The Helicoid (again!)

In 1760, Leonhard Euler studied the curvature of intersections of a surface with planes perpendicular to the surface, and showed that the maximal and minimal values of their curvature are attained along orthogonal curves. In 1776, Jean Baptiste Marie Charles Meusnier de la Place showed that for minimal surfaces these principal curvatures are equal with opposite sign. He went on to show that both the catenoid and the helicoid satisfy this condition, thus exhibiting the first two non-trivial examples of minimal surfaces. Euler had discussed the catenoid as a minimal surface before, but only in the context of surfaces of revolution.

In its standard representation as a ruled surface, the parameter lines are the asymptotic lines of the helicoid. For a change, here is the helicoid parametrized by its curvature lines:


The purpose of this note is a little craft, similar to what I explained earlier using Enneper’s surface: A ruled surface that has as directrix a curvature line of a given surface, and as generators the surface normals, will be flat and can thus be constructed by bending a strip of paper. Doing this for an entire rectangular grid of curvature lines results (for the helicoid) in an attractive object like this one:


To make a paper model, one first needs to find planar isometric copies of the ribbons. This is done by computing the geodesic curvature of the curvature lines of the helicoid, and, using the fundamental theorem of plane curves, then finding a planar curve with the same curvature. The (planar) ribbon is then bounded by parallel curves of this plane curve:

Js 01

Using four (due to the inevitable symmetry of things) copies of the template above, carefully cut out & slit, allows you to easily build the model below, which also makes a nice pendant. Print out the template so that the smallest distance between two slits is not much wider then your fingers, otherwise assembling the pieces will be tricky.

DSC 1166

Begin with the largest J-piece and use the four copies to build a frame, by sliding the hook into hook and non-hook into non-hook. Then continue inwards, adding four copies of the second largest J, by placing the hook of a new J next to a hook of the old J.

DSC 1169

Quadrics in Clay

To get the orthogonal quadrics from Monday into clay using a clay printer, one needs to know about the limitations of Malcolm’s clay printer. It does nothing else but move a vertical tube full of clay horizontally around and vertically up, layer by layer. Simultaneously, it squeezes a continuous stream of clay, with no pause.

DSC 4646

The first few layers are pretty easy, clearly showing the elliptical and hyperbolic cross sections. We only print one half of the whole model, to have a solid foundation (the central cross section), and because it’s cool to be able to look inside.

DSC 4663

Things get interesting when the two branches of the hyperbola come together to connect to the single hyperboloid. We reach a critical point of the height function, and the clay printer clearly has problems with the Morse theory.

DSC 4694

Above you can see the nozzle in action, and more has happened: We have passed a second critical point when the two components of the hyperbola have separated from the ellipse. This is more complicated then the standard Morse theory of manifolds. The printer has do (quickly) move from one component to another at each layer, randomly dropping little chunks of clay on its way.

DSC 4710

This gets a bit messy when we reach the peak of the ellipsoid. Below is the completed print. It needs to dry and be fired. You will notice that we have only used two of the three surfaces. This is a pity, but the missing piece is one sheet of the double hyperboloid, and it is almost horizontal, and impossible to print.

DSC 4741

Arbeit und Struktur

DSC 4006

As hinted at in a previous post, I have been spending a fair amount of time this summer preparing 3D models for clay printing. I will talk about the models and the results at a later point. Today, we focus (or de-focus?) on watching the process. Printing a model takes time (say two hours for a model 20 cm in width) and requires almost permanent attention.

DSC 4140

So one naturally begins to pay attention to details. The shallow focus of a macro lens not only allows to pinpoint these details, it also blurs everything else into pleasant abstraction.

DSC 4309

Color is almost irrelevant, unless one wants to bring out the gradual change of clay type from layer to layer. Everything is reduced to utter simplicity, to the extent that the all too human question for meaning is becoming meaningless.

DSC 4367

What matters is structure, and the work to be done to maintain it.

DSC 3863

Arbeit und Struktur (Work and Structure) is the title of Wolfgang Herrndorf’s Blog-Diary that he wrote in the last three years of his life.
This diary distills much of what mattered to him while facing death, and the title is a further reduction of this to just two words.

Stars and Stripes

A while ago I had the idea for a card game where each card is a square representing both halves of a domino piece simultaneously. That is, each card is decorated with one to six symbols of one kind (say stars), and one to six symbols of another kind (say stripes). Cards could be placed next to each other if they followed the matching rule that requires them to have the same number of stripes or stars. Here is a chain of eight cards, all following the matching rule:

Matching 01

I liked the idea (and I am sure others must have had it before me), but I also had a hard time coming up with a game worthy of this set of cards. Recent developments triggered the much needed idea.

Stars and Stripes is played on a 7 x 7 board. In the initial setup, the 36 cards are shuffled. Each of the 2-4 players draws a card randomly and places it face up into the corner nearest to him or her to mark home. The middle square of the board is occupied by a special card, called the Trump. You can make and decorate it yourself as you see fit, I have kept it gray and empty. Here is how the setup could look like with four players.

Setup 01

The remaining cards are dealt out to all players. If you play with 2, 3 or 4 players, each gets 17, 11 or 8 cards.

The primary purpose of the game is to gather the largest number of followers. A follower is a card on the board that is connected to the player’s home corner through other cards, who are then followers as well.

The players take turns. At each turn, the player must perform exactly one of the following three actions:

  • Place a card from the hand on an empty square of the board that is surrounded in all 9 directions by empty squares or by the border of the board. This card is then called an independent. Placing a card like this can be used to prevent other players to expand their fellowship, or to prepare one’s own future expansion.
  • Place a card from the hand on an empty square of the board so that it borders one or more follower cards of the player, but to no follower card of another player. Cards may border independent cards and/or possibly the Trump.
    Cards must be placed following the matching rule for all neighbors with which they share an edge. This means that the placed card must have the same number of stars or the same number of stripes as each neighbor in the four directions north, east, west, or south. The matching rule is not applied for the Trump. A card placed this way will automatically become a follower of the player, as do all independents this card possibly connects to.
  • Exchange one card randomly with another player. This is done as follows: Both players spread out their cards face down, and both players select simultaneously a card from the other player.

Let’s look at an example. After a few turns, the board might look like this:

Example 01

Player NW (upper left corner) has five followers, NE and SE four, and SW five. There are three independents. NE is blocked by an independent with one star and four stripes. The only way out of it is to play a card with four stripes and six stars or a card with three stripes and one star. This would also convert the independent into a follower.

Let’s suppose it is SW’s turn, and he or she would like to play the card that I placed next to the board. There are only three possible spots left for this card, marked by roman numerals.

By playing in spot I, SW will gain the independent with two stars and stripes as a follower. Playing in spot II just adds the card as a follower, and playing in spot III connects to the Trump.

Only one player can connect with the Trump, and the Trump does not count as a follower. However, by being connected to the Trump, the player is now allowed to break the matching rule:
Whenever he or she wants to place a new card, this card still must be either isolated or only border the player’s own followers and possibly independents, but the card does not need to match in the number of stars or stripes. In other words, the player connected to the Trump has it much easier to increase the number of followers.

The game ends when after a round, a player has run out of cards or no new card has been placed during that round. The winner is the player with the most followers.

For increased fun, this game can be played also on larger boards with several decks of cards.

You can download a pdf file with cards to print and cut out here.
Get it now, while the game is still legal to play.