New Eden (New Harmony IV)

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Nested among a garden of fruit trees next to the Roofless Church in New Harmony is another sculpture by Stephen de Staebler, the Angel of Annunciation, which is easy to overlook, despite its tallness.

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A small plaque on the church wall nearby quotes a poem by Staedler that states that arms are for doing, while wings are for being.DSC 1886

This angel is deeply conflicted. The arm sticks out of his head like the wings. The head itself, whose face is just recognizable as such from the side, is split in half when viewed from the front.

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One of the two feet is cemented in, the other free to walk. Where does this leave us?

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There is another sculpture in this garden, without plaque or any indication of authorship: A piece of wood, hanging from a tree.

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It’s not a sculpture. It’s what is left over from binding the branches of an aging tree together to keep it from breaking and falling apart. An attempt can never completely be a failure. Doing and being can still be one.

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Noli turbare circulos meos! (Annuli VII)

In science, our goal should always be to present with clarity. Since the discovery of perspective drawings, a realistic representation of 3-dimensional objects has become almost mandatory. However, very often these objects have an appeal beyond their scientific truth which gets lost if its is shown in full clarity.

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This blog has two series of posts titled “Spheres” and “Annuli” that both showcase images of simple 3-dimensional mathematical objects which deliberately forsake clarity in order to convey that other appeal. While accurate perspective renderings are used, the  perspective and textures are chosen as to emphasize the abstract aspect. 

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The example above shows a triply orthogonal system of surfaces. An easy way to create such a system is by taking a doubly orthogonal system of curves in the plane, revolve them about a common axis to obtain two families of surfaces of revolution that intersect orthogonally, and add all planes through the axis of revolution. For instance, we can choose two families of touching circles that pass through a common point, as above.

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A single circle, rotated about the black axis, will revolve into a torus. To spice things up, let’s apply an inversion at a sphere centered at the intersection of the circles. This turns the tori into special cyclids like the one above, which all have the appearance of a plane with a handle. Using both a red and a green circle will invert-revolve in two such cyclids that intersect in a straight line and a circle:

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These are still attempts of realistic drawings, but we already get the feeling that things aren’t completely evident anymore. For instance, the two cyclids above should be equals: but where did the corresponding red handle go?

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Above is the same pair of objects from a different perspective. Now we can see the two handles and the intersection in a line, but where is the intersection circle? Also, where do we need to place the third surface family, which consists of inverted planes, i.e. spheres? The answer to that question is indicated below.

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Other perspectives allow amusing variations:

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For the top image, I have used several cyclids from each family, and several spheres, clipping them between two planes. To appreciate the image, all this knowledge might be irrelevant. To create it, it is essential.

 

Fold Me!

Last year, Jiangmei Wu and I worked on some infinite polyhedra that can be folded into two different planes. Today, you get the chance to make your own (finite version of it). This is a simple craft that, time and energy permitting, will be featured at a fundraiser for the WonderLab here in Bloomington. You will need 3 (7 for the large version) sheets of card stock, scissors, a ruler and craft knife for scoring, and plenty of tape. A cup of intellectually satisfying tea will help, as always. 

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Begin by downloading the template, print the first three pages onto card stock, and cut the shapes out as above.  Lightly score the shapes along the dashed and dot-dashed line, and valley and mountain fold along them.  Note that there are lines that switch between mountain and valley folds, but all folds are easy to do.

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The letters come into play next. Tape the edges with the same letters together. Begin with the smaller yellow shape, and complete the two halves of the larger blue shapes, but keep them separate for a moment, like so:

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Stick the yellow piece into one of the blue halves, this time matching the digits. Complete the generation 2 fractal by taping the second blue half to the yellow generation 1 fractal and the other blue half.

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This object can be squeezed together in two different planes. Ideal for people who can’t keep their hands to themselves. 

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The next 2 pages of the template repeat the first three without the markings, if you’d like to build a cleaner model. You then need two printouts of page 5. The last page allows you to add on and build the generation 3 fractal. You need 4 printouts. 

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Cut, score, and fold as shown above.

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Again, tape edges together as before. There are no letters here, but the pattern is the same as before. Finally, wiggle the generation 2 fractal into the new orange frame, as you did before with the yellow piece into the blue piece.

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Here is how they now grow in our backyard. If anybody is willing to make a  generation 4 or higher versions of this, please send images.

All these polyhedra have as boundary  just a simple closed curve. Topologists will enjoy figuring out the genus.

Hidden Simplicity (Maybe-Ferns 5)

Mathematicians like to do things a little differently. An excellent example was the Mathematische Arbeitstagung, a yearly event held in Bonn, where the (mathematical) audience was asked to publicly suggest speakers.

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Friedrich Hirzebruch would write the suggested names on the board (he sometimes misheard…), and then create a list of speakers on the fly. Sometimes they ended up with unexpected results. One year, Michael Barnsley was suggested, who had been working on a new fractal image compression method.

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His talk was exciting for us graduate students, because we for once could understand something. The idea was to use special types of iterated function systems: Take a few linear maps that are all contractions, and use them to map a subset  of the plane to the union of the images of that set under all the linear maps. This becomes a contraction of the space of closed subsets of the plane to itself with respect to the Hausdorff distance, and hence has a fixed point, which is again a subset of the plane.

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It turns out that these subsets are highly complicated fractals, encoded just by a few numbers. For instance, all images on this page (except for the photo of Hirzebruch at the top) were made with just two linear maps, requiring 12 decimal numbers.

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Barnsley claimed that he could reverse engineer this: Start with an image, and find a small collection of linear maps that would produce the given image very accurately. If true, this would revolutionize image compression.  We went home and tried it out on our Atari ST computers and the likes. All we could produce were ferns, twigs, and leaves.

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Paul Bourke has a nice web site where he explains how one can design some simple fractals, and has also some very impressive images of ferns using four and more linear maps. Below are the two simple maps used to create the polypodiopsida psychedelica above.

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Carpets (Foldables 4)

The last (for now) example in this series of bifoldable designs is a woven carpet. Will create a doubly periodic polyhedron that consists of the Miura tubes below (which are almost 50 years old!).MiuraTube

We begin with a corner type we call Double L

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Four copies of it (using reflecions) can be combined into a translationa fundamental piece like so:

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The tubes (of double length) emerge when we replicate this piece several times in both directions:

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Above is its most symmetric state. This carpet does not need to be rolled, it can be squeezed in both of its translational directions, as below:MiuraWeave 4

So you can push this Miura Carpet to any of the four sides of a room.

Dos Equis (Foldables 3)

Whenever you show a mathematician two examples, s(he) wants to know them all. So, after the introductory examples of Butterfly and Fractal it’s time to make something more complicated. Jiangmei and I started by classifying all possible vertex types that can occur when you build polyhedra using only translations of four of the six types of faces of the rhombic dodecahedron (and make sure they attach to each other as they do it there). We found 14 different ones, and a particularly intriguing one is what we called the X:

TripleX1The central vertex has valency 8, and we were wondering whether we could use it to build a triply periodic bifoldable polyhedron. It is easy to combine two such Xs to a Double X:

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One can then put a second such Double X (with the order of the Xs switched) in front. Note that these are still polyhedra. Below are two deformation states of these quadruple Xs. We see that they are quite different.

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So far, the construction can be periodically continued up/down and forward/backward. It is also possible to extend to the left/right, and there are in fact two such possibilities, allowing for infinite variations, because one has this choice for every left/right extension. They are indicated by the arrays below. 

TripleX3carrowIf you don’t have the time to build your own model, here again is a movie showing the unfolding/folding of a rotating Dos Equis.

 

The Fractal (Foldables 2)

The second bifoldable object Jiangmei showed me was this:

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You can find a movie showing how this folds together in two ways here. To understand how and why this works, let’s first look at a simple saddle:

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This is a polyhedron with a non-planar 8-gon as boundary. Its faces are precisely the four types of faces that are allowed in our polyhedra: All others have to be parallel to these four. The four edges that meet at the center of this saddle constitute the star I talked about the last time. Again, all edges that can occur must be parallel to one of these four. One can fold the saddle by moving the upwards pointing star edges further up (or down), and the downwards pointing edges further down (or up), thereby keeping the faces congruent. This works locally everywhere and therefore allows a global folding of anything built that way. Fractal 0

For instance, the hollow rhombic dodecahedron above can be bi-folded. Now note that this piece is also a polyhedron with boundary. In fact, its boundary is exactly the same octagon as the boundary of the saddle. 

Observe also that at the center of this piece we have a vertex in saddle form. This suggests to subdivide all rhombi into four smaller rhombi, remove the saddle an the middle vertex of the doubled hollow dodecahedron, and replace it by a copy of the standard hollow dodecahedron. This gives you Jiangmei’s fractal. Repeating this is now easy. Below is the generation 2 fractal (animation):

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And, just for fun, the generation 10 fractal:

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You can see it being bifolded here. So far, the two completely folded states of our polyhedra looked very much the same. We will see next week that this doesn’t need to be the case.