Zip ties are a great way to tie things together, from cables to people. They are extremely difficult to take apart non-destructively once they have been “zipped”, so you have to be careful when trying to build something out of them, but having a built in connection mechanism makes building things “easy”.
Which is to say that it’s easy to build things as long as you want to make long strips or rings of zip tie, but there is no real way to “branch” your rings. Pentagon? easy. Tetrahedron? hrm…
Fortunately, we don’t need nodes of degree greater than 2 to make cool structures. By interleaving rings of zip ties, we can make compounds of regular polygons. These structures are referred to as regular polylinks or orderly tangles.
Orderly tangles are a nice way to illustrate symmetric colorings. I made this orderly tangle of four zip tie triangles out of four different colors of zip tie. The resulting structure has octahedral symmetry. If you think of the zip ties as representing edges, and the spaces between them as faces, the resulting structure in the middle can be thought of as a woven cuboctahedron.
Since the zip tie triangles have a chirality to them, the final sculpture also has some chirality. Keeping the chirality straight during construction may be the most difficult part of making this piece.
Many other tangles are possible. A tangle of six pentagons is tricky (and I don’t have that many colors of zip tie), but would demonstrate a similar symmetric coloring of the icosahedral group. Let me know if you try making one!
How would one make mathematical cuisine? Not just food that looks mathematical (likemathcookies), but something that you truly have to eat and taste in order to experience its mathematical nature.
Henry Segerman proposed this question, and today we had our first experience tasting the answer. A masterpiece of mathematical art, the answer I came up with is simple enough that anyone can make it at home, surprisingly visually beautiful, delicious…
Layered Drinks!
Not just any layered drinks, of course. In our layered lemonade, the intensity of flavors as you go down the layers increases exponentially. The sugar and lemon juice proportions for the same quantity of liquid increase according to the Fibonacci sequence. The sweeter layers are denser and naturally keep separated lower down. Indeed, the nature of layered drinks is that they are monotonically increasing in sweetness (and/or decreasing in alcohol). Thus, there is an intrinsic mathematical property to all layered drinks.
Additionally, the ratio of sugar to lemon juice in our lemonade isn’t constant. The top layer of our drink is 1 part lemon, while the second layer is 1 part sugar syrup. Following the Fibonacci rule, each subsequent layer has proportions that are the sum of the previous two layers proportions.
Layer 3 is 1 part lemon, 1 part sugar. Layer 4 is 1 part lemon, 2 parts sugar, and so on.
Generically, for layer n > 2, there are fib(n-1) parts sugar and fib(n-2) parts lemon juice. These are adjacent Fibonacci numbers, so as the drink is consumed the ratio of lemon to sugar approximates the Golden Ratio. This drink may be the worlds first tastable example of the relationship between the Fibonacci sequence and the golden ratio!
Surprisingly, using golden ratio relative proportions of lemon juice and simple syrup actually seems to make rather good lemonade. When consumed, the beverage starts out fairly flavorless, then rapidly ramps up. It also alternates between being a bit sweet to being slightly sour (of course this is a matter of personal taste), as the approximation of the golden ratio alternates between being slightly high and slightly low.
You too can make Fibonacci lemonade and experience the taste of exponential flavor, the golden ratio, and the Fibonacci sequence. Just follow the recipe below!
Ingredients and Materials:
Lemon Juice SImple Syrup (1 cup sugar dissolved in 1 cup water)
Water
Food Coloring (optional but makes it pretty)
Ice or a spoon for slow pouring (the ice free strategy is much harder)
The Method
You must start with the sweetest and densest layer and work your way backwards up the drink. I describe the layers for a 7 layer drink below, although you may choose to make a different number of layers to start. The drink could also be made without the first layer, in which case it is neatly just two offset increasing Fibonacci sequences, one per ingredient.
First, fill your glasses with ice. Then, do the following steps for each layer. Finally, sip your mathematical masterpiece.
Add the proportions of lemon juice and simple syrup indicated below to your liquid measuring cup.
Add food coloring if desired.
Fill measuring cup to the 4 oz. (1/2 cup) line.
Stir to blend all ingredients in your measuring cup.
Slowly pour a layer from your measuring cup into your drink glasses. You want to pour directly onto an ice cube, the ice cubes are there to slow down your liquid as it goes down the cup and to help keep the layers distinct. (You can pour the first layer normally)
The Layers
1 tsp. lemon juice
1 tsp. simple syrup
1 tsp. lemon juice, 1 tsp. simple syrup
1 tsp. lemon juice, 2 tsp. simple syrup
2 tsp. lemon juice, 3 tsp. simple syrup
3 tsp. lemon juice, 5 tsp. simple syrup
5 tsp. lemon juice, 8 tsp. simple syrup
Tips
Many simple syrups are 2 parts sugar to 1 part water. If yours is like this, halve the amount of sugar you are using (or it will probably be far too sweet).
For a more authentic, less watered down experience, you need to make your drinks without ice. This is much harder, and I don’t actually recommend it unless you are patient or really know what you are doing. You can find directions for layering without ice here.
Last week, I looked out the window of my office and saw a parrot. Two of them actually. Apparently San Francisco is home to a wild colony of escaped (released?) Cherry Headed Conures. There is even a documentary about them, but somehow I had seen nor heard of them before.
I promise, these birds were parrots.
Walking back to my desk, I had a moment of inspiration.
The internet is kind of amazing, and I quickly found a website that would sell me parrots in 3 different colors for 39 cents apiece. I think they are supposed to be used as accents for floral centerpieces; I have no idea how anyone is making a profit selling them at that price.
A parrotohedron obviously needs to be pyritohedral, so I started by arranging six sticks pyritohedrally for my parrots to perch on. Then, I arranged 12 parrots (one per stick end) such that each parrot had a friend (and represented one face of a pyritohedron). Squawk! It’s a Polly-hedron.
But will there be a pirate-ohedron? Only time will tell.
ps. Arrr. Where be my two-dimensional parrot?
pps. Polly gone.
First, lets talk a little about 2D hyberbolic tilings for those who haven’t encountered them in the past. On a Euclidean plane, we can tile squares onto a checkerboard such that four corners meet at a point.
A boring Euclidean chess board is a tiling of squares where 4 squares meet at each point.
On a hyperbolic checkerboard more than 4 regular quadrilaterals meet at a vertex when tiling a plane. This means each angle of a square must be less than 90 degrees (if 5 squares meet at a vertex in a planar tiling, then the angles should be 360/5 or 72 degrees).
There are lots of potential regular hyperbolic tilings that we could imagine playing chess on. We’d like to preserve the checkerboard property of our tiling, so we will only consider tilings where an even number of regular polygons meet at a vertex. Here are a few potential tiling candidates drawn using the Poincaré disk model*.
The {4,6} hyperbolic tiling is a tiling of 4-sided polygons (squares) such that 6 of them meet at a vertex. It’s nice because it keeps the shape of the tiles in the checkerboard as squares.
{4, 6} Regular Hyperbolic Checkerboard
A {5, 4} tiling where 4 pentagons meet at a vertex maintains the property that a piece moving along the diagonal can only visit tiles of one of the two colors. On the other hand, it becomes extremely difficult to define the opposite side from a side or the opposite corner from a corner, so moving in a straight line quickly gets messy.
{5, 4} Regular Hyperbolic Checkerboard
Finally, the {6, 4} and {4, 8} tilings start to get pretty busy, but do nicely maintain the idea of having opposite edges well-defined (lacking in a pentagonal tiling), while keeping opposite corners colored the same.
The first issue that we face with our hyperbolic chess board is that it isn’t going to lie flat in 2D Euclidean space, so pieces are going to want to slide off (thanks, gravity!). Thus, playing hyperbolic chess with a physical board requires either that our pieces actually stick to our board (velcro?), that we put up with the large amount of square distortion that occurs when trying to embed a hyperbolic board on a Euclidean plane, or that we write and play our game on a computer.
Looking at the tiling candidates above, the {4, 6} tiling and other square tilings seem like clear winners for our hyperbolic chess game. In particular, it is immediately obvious what it means to travel horizontally and vertically across it.
Examples of rook (blue) and knight (green) moves
The main question is what it means to travel diagonally across the tiling, when leaving a corner of the {4,6} tiling there is one obviously diagonal square, and two still kind of diagonal squares adjacent to it. However, the obviously diagonal square doesn’t share a color with the original square, which means that we lose one of the standard rules of chess – that each bishop can only visit a single color of square. It also more or less ruins the game of checkers, in my opinion. Moreover, if we only allow the “obvious” diagonal, then a bishop would be unable to reach a majority of the squares in the tiling. Therefore, I instead define a ‘reasonable diagonal’ to a square, A, as any square, B, that shares a vertex and a color to A.
For checkers it seems reasonable to allow going to either of them at every diagonal. This seems to be in keeping with the spirit of the game and to be likely to keep game play reasonably balanced.
It’s less clear that this is a good strategy for chess. It certainly makes the pieces that can travel along diagonals much more powerful, as they can now travel to many more squares than ones that can only travel horizontally or vertically. That said, my guess is that a reasonable strategy would be for pieces moving along the diagonal to be able to make a choice when moving out the very first corner to either take the “right” or “left” path and to always take that path on any given move. Thanks to the negative curvature of the board and the straight lines we are requiring, this does leave an interestingly large space that it would take any given piece multiple turns to reach.
An example of a diagonal “bishop” move
Another question is how big to make the surface and where to position the pieces at the start of the game.
Since the horizontal and vertical distances are pretty well defined, we can use them to define the size of the board and the location of the pieces. We start by placing the White queen on a white square. Place a white pawn in a vertically adjacent square. place the remainder of the white pawns such that they are in the same horizontal line as the white pawn you have just placed (with the same number on each side as in chess, of course). Place the remainder of the white pieces such that they are placed correctly as in chess and in the same horizontal line as the white queen. Define one edge of the board as directly behind these pieces.
From the white queen count 7 more squares vertically in the direction of the white pawns, and place the black queen there. Following similar rules to the above, place the remainder of the black pieces and define another edge of the board.
It would be nice to define sides to our board. Unfortunately, this is necessarily not as nice or easy to do as the front and back the way that we have defined the rest of the board. In chess, sides are most important for queening, which only requires the back edge, and for cornering the pieces (ie. the King during endgame). This means that defining sides isn’t strictly necessary, but we could possibly define them anyways by starting from the corner piece and constantly turning towards the center of the board to define side edges.
Pentagons are my favorite polygon, so I’m pretty fond of the {5, 4} tiling. If chess had originally been developed on a board like this, we might imagine that the standard move might involve entering a square on a face and leaving at the opposite vertex or vice versa. Perhaps rooks would have to start their move going across an edge, bishops would have to start going across a vertex, and queens could do either. Knights have the natural travel rule of having to go through an edge, then through one of the two far edges. I love the elegance and naturalness of these rules for the tiling. One might almost imagine that chess was invented for this tiling and that our Euclidean version came out of an attempt to modify the rules to fit a Euclidean chess board.
Examples of moves that a rook (blue arrows) or a knight (pink arrows) could make on this board
I hope you’ve enjoyed our adventures through non-Euclidean chess. It’s interesting to realize how much the apparently** Euclidean nature of our world influences the games that we play and the rules that we make for them. Rules that are straightforward in Euclidean geometry become bizarre, insufficiently well-defined or otherwise sub-optimal in other geometries and vice-versa.
* Hyperbolic tilings were created with the aid of KaleidoTile ** According to general relativity, space is curved, but we don’t notice it so much at our scale
I first posted about an origami model based on a Star Polyhedron in November 2012, but I knew at the time that many more models were possible based off of a similar idea of taking a Star Polyhedron and removing parts of each face such that some underlying topology of the polyhedra was maintained, but the faces no longer intersect. These representation of a star polyhedron can be thought of as ‘topological’ as they emphasize the internal connectedness of these self-intersecting figures.
Unukalhai, below, is a ‘topological’ model based on the Small Triambic Icosahedron and is composed of 60 rectangular sheets of paper folded into identical origami units. Each face is represented by a 3-pronged spiral, and can be thought of as a subsection of a small triambic icosahedron, chosen such that the model can be joined together without self-intersection. Unukalhai was made out of five colors of paper to highlight the natural five-coloring of the icosahedron.
Tania Australis was the first star polyhedra origami piece that I folded, and has been mentioned on this blog before. It is composed of 30 identical ‘S’ shaped pieces that each represent one face of a Great Rhombic Triacontahedron and was inspired by George Hart’s Frabjous sculpture.
Tania Borealis is composed of 30 ‘S’ shaped pieces, put together in the shape of a Medial Rhombic Triacontahedron. It complements Tania Australis, but is much more fragile, so I do not plan to put it on display in Asia.
Squawk! It’s a Polly-hedron.
