Art and Science - Hyperbolic Blanket

A mathematician friend of mine once told me that "math is the study of interesting definitions." Mathematicians, like them, study abstract systems built up from starting assumptions, known as axioms, and attempt to discover new features of these systems through the logic of formal proofs. The choice of foundational axioms is incredibly important to the resulting systems, as a slight change to one of these building blocks can create an entirely new world full of rich structure. People who do math make their choice of axioms by considering what new, useful, or interesting structures arise from those axioms.

Non-Euclidean geometry perfectly encapsulates this exciting feature of mathematics. Last week I had the pleasure of attending a lecture by Dr. Evelyn Lamb about hyperbolic geometry--in particular, visualizing its counterintuitive properties. Hyperbolic geometry is a type of Non-Euclidean geometry that was discovered by changing one of the five fundamental axioms chosen by Euclid over two thousand years ago. The axiom in question, called the parallel postulate, states that, given a line and a point off that line, there is a single, unique line that passes through that point and never intersects the first line, no matter how far the two lines are extended. Change this assumption, and the shape of your space is dramatically altered. 

Euclidean geometry is the geometry of the flat plane, the geometry I was taught in middle school, the geometry you can draw on a piece of paper. The consequences of choosing Euclid's version of the parallel postulate have been explored for thousands of years. From it we get results like the Pythagorean Theorem and the theorem that the interior angles of a triangle add up to 180 degrees, among others. 

So what happens when we change the parallel postulate? Let's rephrase it: given a line and a point off that line, there is NO parallel line that passes through that point that will never intersect the first line. When extended, all lines eventually intersect. This causes the space to close up on itself and gives us spherical geometry, which is familiar to us when we look at a globe. We live on the two-dimensional surface of a sphere, and this axiom system describes it. In this geometry, we get counterintuitive results. For instance, interior angles of a triangle add up to more than 180 degrees.

We can change the parallel postulate again. Here's the third version:  given a line and a point off that line, there an infinite number of lines that passes through that point and they never intersect the first line, no matter how far the lines are extended. This is hyperbolic geometry, an even stranger structure that is difficult to describe without the language of mathematics. It is exponentially expansive, and just like a sphere, it is impossible to draw in a Euclidean plane without distortions. This axiom system gives us triangles with interior angle sums that are less that 180 degrees. In fact, in hyperbolic geometry, it is possible to have triangles that have interior angle sums of zero!

Dr. Lamb gave us a few techniques for conceptualizing the hyperbolic plane in the talk, from crochet to 3D printed models. But what captured my imagination most was a pattern for a hyperbolic blanket. So I decided to make it.

How can you translate hyperbolic geometry into the surface of a physical blanket? Using tessellation, or tiling. Just as equilateral hexagons, squares, and triangles can tile the Euclidean plane, other shapes can tile hyperbolic and spherical surfaces. 

In the Euclidean plane, we have 360 degrees of space around every point. A tiling of equilateral triangles demonstrates this. At each point, six equilateral triangles meet, each with all interior angles equal to 60 degrees. Multiplying 6 by 60 yields 360, as expected. The picture below demonstrates this tiling, made from toy magnets.

In spherical geometry, there are less the 360 degrees of space around each point. This allows the surface to close into a bounded spherical surface. The tiling below shows five equilateral triangles meeting at each point, each with all interior angles equal to 60 degrees (although in the flat picture, they don't look equilateral due to distortion, you'll have to trust me). Multiplying 5 by 60 gives us 300, much less than the expected Euclidean 360. See below.

Finally, in hyperbolic geometry, there are more that 360 degrees of space around each point. This makes the surface expansive and very floppy. The tiling below shows seven equilateral triangles meeting at each point, each with all interior angles equal to 60 degrees (again, they don't look equilateral due to the pesky distortion). Multiplying 7 by 60 gives us 420, more than the expected Euclidean 360. Here's a picture.

The blanket pattern I used uses a tiling with four pentagons clustered around each point. The interior angles of a pentagon are 108 degrees, so fitting them four-to-a-point gives us 432 degrees to accommodate. The only surface that can do this is hyperbolic. The cozy result is shown below.

There is so much more to explore when it comes to hyperbolic geometry, and much of it is available online. For more art featuring hyperbolic tilings, look to the works of M. C. Escher. For some mathematics-based intuition, I recommend this series of Numberphile videos that explore what it would like to live on a hyperbolic surface.

Exploring the Cuboctahedron

I built a math toy!

This particular object is in the shape of a cuboctahedron--an Archimedian solid that is particularly fun to play with. With twenty-four bits of straw and a long piece of string, you can build one of your own and morph it into various shapes yourself. The task of building a cuboctahedron incidentally involves learning a bit of geometry and graph theory, along the way.

The cuboctahedron is an Archimedian solid, and knowing what these solids are can actually help you build one. If you look at the corners (called vertices--in orange and yellow below) of a cuboctahedron, you'll notice two square and two triangular faces meet at each vertex. Additionally, the triangles and squares alternate such that two square (or triangular) faces are always opposite each other on the vertex. No two of the same shapes share a side. This pattern is the same at every vertex, as is true for any Archimedian solid. If you build one vertex with this pattern, then continue it with each new vertex you add, you will eventually complete the cuboctahedron correctly. I enjoy making cuboctohedrons and other geometric objects in this algorithmic way, because instead of comparing the model in my hand to a reference on paper, I can use logic to reason what my next steps should be and avoid a lot of confusion that results from comparing a three dimensional object with its two dimensional representation.

The cuboctahedron I made uses straws for its edges. The cuboctahedron has twenty-four edges, so if you want to build it, you'll need twenty-four identical straw pieces and a string at least as long as all of the straw pieces combined. The string only needs to be in one continuous piece--it's possible to wrap the string through each straw, passing through each straw segment once and only once, with the end of the string finishing at the same place it started. This way, you can knot the two ends of the string together and have a flexible cuboctahedron with minimal knotting.

The type of string path we want around the edges of the cuboctahedron, a path that starts and ends in the same place after passing through each edge only once, is called an Eulerian circuit. It's only possible to complete on the edges of solids if an even number of edges meet at each vertex. It's easy to see why--in order to have a complete cycle, at every vertex, the path of the circuit must both enter, and then leave that vertex. Since each vertex can only be traversed once, each "entering" path must be paired with an "exiting" path. If you don't believe me, you can try building solids that don't satisfy this property with only one single piece of string. Mathematics guarantees you will inevitably fail, though you can still build these solids with the less nice more-than-one-string-required property if you want.

Here is the Eulerian circuit I used in my cuboctahedron. There are other ways of "lacing up" the toy, too! At each vertex, I wrapped the string around itself a few times so that all four edges would hang together nicely. This also let me space out the ends of the straws, and gave the toy a little bit more flexibility (a bit of engineering and experimentation helps).

Art and Science - Symmetry Poems

The idea behind symmetry is a simple one: what operations can you perform on an object while preserving its appearance or structure? From this simple question, an infinite variety of possible patterns emerge--as the abundant symmetries found in mathematics, art, and nature can attest to. In order to explore the possibilities of symmetry in poetry in a more manageable way, we need to restrict our focus to a certain class of symmetries known as frieze patterns.

Frieze patterns are symmetrical patterns that extend infinitely along one direction, like a number line. Every frieze pattern contains at least one symmetry: translational symmetry, which guarantees that the pattern repeats in space after a finite distance. Shifting the entire pattern by integer multiples of this distance preserves the structure. In addition to translational symmetry, frieze patterns can also contain reflections about horizontal and vertical lines, as well as 180º rotations and glide reflections about horizontal lines (glide reflections are simply reflections combined with translations). An alternative way to visualize frieze patterns is to imagine building them. Start with a simple, asymmetrical shape to use as a seed, and then transform it using rotations, reflections, translations, and glide reflections as needed. Amazingly, with these four types of transformations, it's only possible to build seven distinct symmetric structures out of the same seed.¹ These seven patterns are referred to as symmetry groups, and frieze patterns represent a specific type of structure called a two dimensional line group.²

In an earlier post, I created a graphic representation of the pattern found in a poetic form called the sestina. A few weeks later, I was experimenting with writing a pantoum, a poetic form I had never tried before. I created a similar graphic in order to visualize the repetition pattern and it reminded me of frieze patterns I had seen before. I decided to explore symmetries in poetry by starting with a simple seed that links two lines in different stanzas. Then, I colored each frieze pattern in order to group together lines that were similar.

Two similar lines might:

• Rhyme with each other, as in a sonnet,
• Contain the same number of syllables, as in a limerick or ballad,
• Have the same first word or last word, as in a sestina, or
• Be the exact same line, as in a pantoum or villanelle!

Below are the poetic forms I found for each symmetry group, along with the name of the pattern the poem is based off of and a color and letter coded similarity scheme for each stanza (two blue a's, for instance, represent lines that are similar to each other in some way). Feel free to write your own poems using these styles, or modify the patterns to make up new styles as you see fit. For instance, I like to preserve the rhyme pattern all the way to the end, and then repeat the rhymes I used at the beginning so the poem is cyclical. I haven't tried most of these forms yet, so I don't know which ones work the best! I would love to hear about any discoveries you make.

Hop: A four-line³ sestina.⁴

Step:

Sidle:

Spinning Hop: A pantoum arranged into four line stanzas.

Spinning Sidle:

Jump: A terza rima. 

Spinning Jump:

***

¹ A fun seed to start with is the shape of your footprint. The names of the symmetry groups might give you a hint on how to produce each pattern. 

² For more on the mathematics of symmetry and group theory, I recommend this book.

³ Of course, a sestina doesn't necessarily need to have four lines...

⁴ There are lots of possibilities for a "hop" type poem, because it contains the least symmetry--it simply contains translational symmetry. 

Art and Science - Sestina Numbers

Earlier this year, I wrote a sestina which was then published in Totem, Caltech's literary magazine. Sestinas are one of my favorite types of poems, because they use a complicated repetition scheme that gives structure to the poem without using rhyme. Knowing this, one of my friends pointed out to me a generalization of the structure of sestinas that produces a sequence of numbers with interesting and surprising properties.

Sestinas have 39 lines each, with six stanzas and a three line ending. Ignoring the last three lines, each of the six stanzas use the same six words at the end of each line. This is what makes writing a sestina difficult--you have to pick six versatile words and avoid repeating the same message in each stanza! However, in each stanza, the six words are in different orders. The pattern by which the words get shuffled to produce the next stanza is the same between each stanza: the last word of the previous stanza is always the end word for the first line of the next. By the end of the poem, five shuffles later, repeating the shuffling procedure will produce the original order of the six words. Here's a picture of how it works:

Mathematicians call this reshuffling a permutation. If you've been looking closely, you might have noticed how it works for this particular permutation. To to figure out where the n^th word goes, first figure out if n is in the first half or second half of the stanza. For n in the first half, the n^th word will be the 2n^th word in the next stanza. For n in the second half, the n^th word will be in the 2*(6-n)+1^th position. The paper linked to below suggests a good way to think of the permutation: as a shuffle (alternating between the first three numbers and the second three numbers) with the second group of  "cards' turned upside down (so that the last word becomes the first word). This type of permutation can be generalized for any number m stanzas, but the number will only be a sestina number if after m permutations, the original order is obtained. 

The picture below shows a braid pattern that represents each permutation. The black lines designate the final word ordering for each stanza. Notice how, if you wrapped the picture around on itself, the colors would connect to each other in the same order as they started (representing the order of the first stanza). For knot theorists, this means sestinas form links with six loops. 

As it turns out, sestina numbers have a lot of interesting mathematical properties related to prime numbers. For instance, if s is a sestina number, then 2s+1 is a prime number! Many sestina numbers are also prime numbers, too. One method for proving an infinite number of sestina numbers exist depends on the truth of the Reimann Hypothesis, an unsolved problem in mathematics that is deeply connected to the distribution of prime numbers. Unexpectedly, sestina numbers are related to both beautiful poetry and beautiful mathematics. 

Read more about sestina numbers (also called Queneau Numbers) here. 

Read more about sestinas here. 

Art and Science - Glassblowing

This week, I had the opportunity to visit a glassblowing studio. I discovered there was a lot of science involved in glassblowing, which seemed at first like a solidly artistic endeavor. Materials science in particular is important to understanding how glass will react in various circumstances, and knowing how materials respond to different conditions is vital to having control over the work being produced. For example, hot glass doesn't stick to cool steel, so in order to gather a blob of glass, you need to first heat a metal rod. Cold metal, however, works well as a surface for shaping glass. Fluid dynamics is also important. Molten glass turns out to have a honey-like consistency. It flows, but very slowly, and it droops in response to gravity if held still for too long. To prevent the glass from dripping onto the floor, the rod needs to be turned at all times.

A rather dramatic example of how science becomes relevant to glassblowing is the Prince Rupert's drop. Imagine taking a blob of glass and letting it drop into a bucket of water. The glass ends up forming an elongated teardrop shape with a very thin, twisted tail. These tails can be as thin as a human hair. Due tension created in the glass during rapid cooling, the bulb of the drop in incredibly strong. Smash it with a hammer, and it will not burst. However, breaking the thin tail causes the entire drop to burst, creating a fine white powder of glass particles.

I ended up making a heart-shaped paperweight. I still can't come up with a good metaphor for what it felt like to shape it that captures both the heat and malleability of molten glass.

We often tend to think of the craftsperson, painter, or sculptor as purely an artist. The skills for creating these media are acquired over years of experience. Over time, the artist tests different methods of creating a work in order to find out which techniques are the most effective. After a lifetime, this built up knowledge is vast, allowing an artist to deal with almost any situation they encounter. This body of knowledge, acquired empirically, is just like the body of knowledge most people imagine when they think of science. The artist is doing science when they discover a new method for working with their chosen material. By repeating the process over and over again, they can test the reliability of the effect and build a style. And when something goes wrong, the artist instinctively checks for what happened during the process, seeking out variables that changed the result of their experiment. An artist develops theories: cool glass quickly and it is fragile, like the Prince Rupert's drop, but cool it slowly and it is strong. Certain colors, when paired together or treated incorrectly change state and produce unexpected hues. The method used by artists to create and explore new techniques is science, and there is a lot we can learn from these artists' experiences.