# Five Intersecting Tetrahedra

One of my favorite mathematical models is the Five Intersecting Tetrhadra model. I first became aware of this model from Thomas Hull, a mathematician at Western New England University. He has written extensively about mathematics and origami (check out his book Project Origami). In particular he has written fantastic instructions for creating this model using modular origami.

The five intersecting tetrahedra model is based on the dodecahedron. This one of the five classic regular polyhedra consisting of 12 pentagonal faces and 20 vertices. Take 4 vertices in the dodecahedron which are the same distance apart. These form the 4 vertices of a regular tetrahedron, as shown on the right (figure from Tom). Since the dodecahedron has 20 vertices, we can inscribe 5 such tetrahedra inside the dodecahedron.

The origami version comes about by making a thin frame for each tetrahedron. Provided the frames are thin enough, they won’t intersect each other. Instead they will form a marvelous interwoven pattern. As shown on the left, I have successfully followed Tom’s instructions for making the origami version of this model.

The next natural question is whether or not the model can be 3D-printed. The answer, is of course, yes! I used crsfdr’s model Interlocking Pyramids  from Thingiverse to print the model on the UPrint SE printer at W&L.  The photo at the top shows the finished version. The one to the right shows the model just out of the printer before the supports have been removed. Since the supports are made of a material that dissolves, the UPrint was the perfect printer for the job.

My colleague Professor Michael Bush has used the Five Intersecting Tetrahedra model when teaching introductory Group Theory to undergraduate students. Indeed, this was the motivation for 3D-printing the model as the origami version is not really robust enough to use in a classroom setting. The model is a great tool for discussing the rotational symmetries of the dodecahedron (or its dual the icosahedron). Roughly speaking, the rotational symmetries of the dodecahedron act in a natural way on the five tetrahedra giving a permutation representation of the symmetry group. After some noodling around this allows one to see that this group is the alternating group $$A_5$$. (Michael usually suppresses details about the faithfulness of the representation at this level.)

# Crocheted Hyperbolic Planes

Two posts ago, I described my wonderful visit to the Double Helix STEAM school in Athens GA. My goal was to explore hyperbolic planes with the students. They can be hard to understand and visualize, so I thought I’d write another post with some further musings.

Take a piece of paper, and place the it on a flat surface like a table (or wall or board, or something). Notice that each point on the paper is above exactly one point on the table. Now take a ball (or something round – like the top of your head). Try do do the same thing with the paper. What do you notice?

You will notice that the paper needs to be creased in order for it lie smoothly against the ball. If you want to have exactly one point on the paper above one point on the ball, then you will need to remove some paper. The crocheted hyperbolic plane shown above is the exact opposite. There is “too much” plane for the paper. In order to have exactly one point on the paper above one point on the hyperbolic plane, then more paper needs to be added.

It is this idea of removing and adding paper that allows us to construct  paper models which approximate spheres and hyperbolic planes. The plane can be thought of as being tiled by hexagons, like a honeycomb. Now, replace one of the hexagons with a pentagon. If you keep doing this in the right way, you’ll end up with a soccer ball shown on the left below. There, each pentagon is surrounded by hexagons. Since hexagons have been “removed” (in comparison with the planar honeycomb), the surface curves inwards and around creating a spherical shape. If, on the other hand, you replace a pentagon with a heptagon (7-sides) and surround it by hexagons, then “extra” hexagons have been added.  The surface will open up and will approximate a hyperbolic plane, as shown below on the right.

The mathematical notion of intrinsic curvature is what lies behind these shapes. Spheres have positive curvature, planes are flat (zero curvature), and hyperbolic planes have negative curvature.  Many people have written wonderful explanations of these ideas. I’d like to just focus on just one consequence. At the end of this post I’ve included a list of resources where you can read more.

Let’s start by thinking about flat surfaces and traditional Euclidean geometry which we all learned (and forgot!) in school. There we learned that the angle sum of the interior angles of a triangle is $$180^\circ$$. That is $$\angle A+ \angle B + \angle C=180^\circ$$, as shown on the right in the figure below. In order to understand what happens on other surfaces, we need to understand triangles. These are geometric figures where three points, the vertices, have been joined by three straight-lines or geodesics.

On the sphere, it turns out that “straight lines”, or geodesic lines, are sub-arcs of great circles. These are circles formed by the intersection of the sphere with a plane through the center of the sphere. For example, if we think of the earth as a sphere, then the equator and lines of longitudes are all examples of geodesic lines. However the other lines of latitude are not geodesic lines, since they are the intersection of the sphere/earth with planes that do not go through the center of the sphere. In the figure above, you can see a spherical triangle with one side part-way around the equator and the other two from the north pole to the equator. Angles $$\angle A$$ and $$\angle C$$ are both $$90^\circ$$, and $$\angle B$$ is also $$90^\circ$$. This means that $$\angle A + \angle B + \angle C = 270^\circ>180^\circ$$! Indeed, for any spherical triangle the sum of the interior angles is bigger than $$180^\circ$$, that is $$\angle A + \angle B + \angle C >180^\circ$$.

Can you see that the red spherical triangle bounds two areas? (One smaller obvious one, and the larger one on the rest of the sphere.) Can you work out what is the largest possible sum of interior angles of a spherical triangle? It turns out that small spherical triangles approximate planar triangles, so their interior angle sum is close to, but still greater than $$180^\circ$$.

What about hyperbolic triangles? Turns out you can create them on a  crochet model by sewing straight lines for the triangle sides, as shown on the left.  Hyperbolic triangles are “skinny”; the sum of the interior angles is less than $$180^\circ$$. In the triangle shown below, you can clearly see the interior angles are small and their sum is much less than $$180^\circ$$. As in the spherical case, small hyperbolic triangles are approximately planar; here the interior angle sum is close to, but still smaller than $$180^\circ$$. Amazingly, the area of hyperbolic triangles whose vertices approach infinity is finite. The proof of this result can be found in any undergraduate textbook on hyperbolic geometry.

I promised to include some references, and here they are.

Two of my favorite undergraduate texts include:

Post Script: When crocheting these hyperbolic planes I weighed out three balls of the same yarn in different colors. Since the length of yarn was approximately the same, the number of crochet stitches were approximately the same, and hence the area made was the same.  Pretty amazing to think that the blue area and the green area around the edge are the same! It just shows how much extra fabric is created as the hyperbolic plane is crocheted.

# Flowers everywhere

In early May, the IQ center at W&L was filled with 3D-printed flowers. Dave Pfaff and his work study students printed flowers in many bright colors on the Cura 3D printer. They are from Super Flowers found on Thingiverse. The fine filaments are created by printing in the air. That’s right, the printer puts down a single layer of material, then returns to the center. Since there are no supports underneath, a “droop-loop” of filament is created.

The vases were also designed by Dave Pfaff. He started with a flower shape on the base, then expanded and twisted the shape around creating the vase shape we see. Fabulous work!

# A new challenge

In December, I was contacted by Professor Ricardo Nemirovsky from San Diego State University to design 3D printable surfaces for the  Taping Shape* exhibit at the Rueben H. Fleet Science Center in San Diego, California. The exhibit runs from January 30 through June 12, 2016.

The exhibit contains a structure made out of packing tape with
three interconnected regions: a torus, a topological
equivalent to Schwarz P surface, and a pair-of-pants
surface with the legs twisted. The structure is large enough for visitors to walk and crawl through. There are three “work tables” (one for each region), with materials, suggested activities, poster displays, etc. The 3D printed models will be a part of the work table and displays.

Ricardo requested I make pair-of-pants surfaces with caps that can be joined together in different ways, Schwarz P surfaces that can be joined together, and also a frame that allows the Schwarz P surface to be created as a soap film spanning the frame. The challenge was on!

In the following blog posts, I’ll explain a bit about the math behind the surfaces, and how we figured out how to build and print them.

*The Taping shape exhibit is part of the InforMath project funded by the National Science Foundation (DRL-1323587).  (The InforMath Project is a partnership between San Diego State University and several museums at the Balboa Park, including the Rueben H. Fleet Science Center .)

# Decorative knots

I’ve been back printing on the FormLabs 1+ liquid printer this week. I decided it was time to print out some more knots, this time for use as festive decorations. I printed out Laura Taalman’s Trefoil Managerie and used some red gift ribbon to create ornaments. Previously Laura had given me a torus with a (5,2) torus knot. I also turned this into a decoration.

I’ve also printed more snowflakes. Again from Laura Taalmans’ Snowflake Machine, the small snowflakes that have a circle so you can hang them up. Awesome.

# Math is all around us…

The holiday’s are fast approaching, and I had a flash back to the movie Love Actually and the song “Love is all around you”. I feel the need to change the first word here (Math = Love?) , since many Christmas decorations show a remarkable amount of geometry to them.

I went to this wonderful tree lighting ceremony and cookie party at W&L and was completely distracted by both the amount of sugar available (cookies, cupcakes, brittle, candy, cider, hot chocolate, …) and the fabulous decorations. These included the stellated star (above) and the fabulous minimal surfaces created by stretching fabric between a table top and it’s legs. The red lighting just added extra pizzazz.

I’ve also seen a lot of knots out there in the real world. My favorite is this folded trefoil knot on the back of a truck. Exactly the kinds of knots I’ve been researching with my undergraduate students.

One of my friends I knit with has the best earrings – nice and knotty! Fun way to be festive at this time of year. (She’s a graphic designer, so was greatly amused at my need to take a picture of her earrings!) I believe that this earring is a true lover’s knot.

# Snowflakes

I’ve a had blast the past week or so 3d printing some snowflakes. They are for the math department for decorations around the holidays.

I’ve been using Mathgrrl’s (aka Laura Taalman) snowflake ornaments (Thigiverse thing 195032). She’s since customized the design here: http://www.thingiverse.com/thing:570339.

I also had a blast printing some of her fractal snowflakes from the Snowflakerator snowflake machine found here: http://www.thingiverse.com/thing:1159436.

I even designed a few of my own found here, here and here.

# Math at the Simon’s Center for Geometry and Physics

I had the very great privilege of being a co-organizer of a workshop held at the Simon’s Center for Geometry and Physics and NYU Stony Brook. This was the workshop on the Symplectic and Algebraic Geometry in the Statistical Physics of Polymers. It was my first time to this campus, and I had a blast with both the math at the workshop AND all the visualization of math in the environment.

My first hint that things were going to be special, was the fantastic Umbilic Torus sculpture found at the end of an avenue of trees between the center and the math department.

The sculpture is by Dr Helamun Ferguson, click  here to find a photo gallery showing the design and construction of the piece.

The sculpture consists of a space filling curve all over the surface of the sculpture. The sharp curve along the edges is a trefoil knot, winding three times around the central hole (the longitude on the torus) and twice around the sculpture the other way (the meridian on the torus).

The base of the sculpture is a large round granite disk with a 3 sided deltoid mirroring the 3-fold symmetry of the sculpture overhead. The base had to be left to settle for a year, and was greatly loved by the local skate-boarders!

The Simon’s Center itself is in a wonderful airy building, with mathematical themes blended seamlessly in the design. I kept finding treasures as the workshop went on. The most obvious, is the sandstone wall behind the stair case leading up to the cafe on the second floor. It is covered with small math motifs from knots, to physics, to finding the square root of 2.

Even the screens on the side of the first floor lounge are mathematical, with different tilings of the plane illustrated. Just love the artistry of the designs in them.