By Angel Abney, Andy Tyminski, and Pawel Nazarewicz

Mathematics is all around us. As we discover more and more about our environment and our surroundings we see that nature can be described mathematically. The beauty of a flower, the majesty of a tree, even the rocks upon which we walk can exhibit nature's sense of symmetry. Although there are other examples to be found in crystallography or even at a microscopic level of nature, we have chosen representations within objects in our field of view that exhibit many different types of symmetry.

This semester in transformational geometry has altered our views, or at least our viewfinders. It seems that everywhere we look now our eyes are drawn first to the patterns of symmetry that exist, and that the object itself is a secondary consideration. So, come join us as we examine examples of bilateral symmetry, radial symmetry, strip patterns and wallpaper patterns in nature. You'll never look at your world the same way again...

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Click on any of the topics below to explore how that symmetry type can be found in nature:

 Radial symmetry is rotational symmetry around a fixed point known as the center. Radial symmetry can be classified as either cyclic or dihedral. Cyclic symmetries are represented with the notation Cn, where n is the number of rotations. Each rotation will have an angle of 360/n. For example, an object having C3 symmetry would have three rotations of 120 degrees. Dihedral symmetries differ from cyclic ones in that they have reflection symmetries in addition to rotational symmetry. Dihedral symmetries are represented with the notation Dn where n represents the number of rotations, as well as the number of reflection mirrors present. Each rotation angle will be equal to 360/n degrees and the angle between each mirror will be 180/n degrees. An object with D4 symmetry would have four rotations, each of 90 degrees, and four reflection mirrors, with each angle between them being 45 degrees.

 A starfish provides us with a Dihedral 5 symmetry. Not only do we have five rotations of 72 degrees each, but we also have five lines of reflection. Another example of a starfish - as we can see, starfish can be embeded in a pentagon, which can then be connected to the Golden Ratio ... Also found in the sea are sand dollars. They too, have D5 symmetry. Jellyfish have D4 symmetry - four rotations of 90 degrees each. It also has four lines of symmetry, and in the middle you have a four-leafed clover for good luck. Flowers offer a variety of radial symmetry. Hibiscus - C5 symmetry. The petals overlap, so the symmetry might not be readily seen. It will be upon closer examination though. This hibiscus is slowly wilting away, and the C5 symmetry is really evident. After some debate, we decided that this flower has C4 symmetry - one row of the petals is underneath another. Petunias offer some patriotic D5 symmetry.

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 Strip pattern symmetry can be classified in seven distinct patterns. Each pattern contains all or some of the following types of symmetry: Translation symmetry, Horizontal mirror symmetry, Vertical mirror symmetry, Rotational symmetry, or Glide reflection symmetry. The seven types are T, TR, TV, TG, TRVG, TGH, and TRGHV.

 An Eastern White Pine has interesting symmetry on it's trunk. Each year, as the tree grows, it develops a new ring of branches (most of which have been broken off in the picture above). The rings move up by similiar translation vectors, but some variation occurs due to the conditions for that year. Another picture of the white pine - this time with branches showing. The white pine exhibits T symmetry. Here is a set of animal tracks that exhibits THG symmetry. Because we have a horizontal mirror, we get a glide by default. I think that these are armadillo tracks.

 Here are some footprints on the beach - in them, we can see a translation and nothing else. That pattern is a typical T. I'm not sure how this pattern was accompished, but it looks like whoever did it had two left feet ... Here is a set of radioactive footprints creating a TG pattern. John Conway simply calls this "step".

 The copperhead is one of the four poisonous snakes in the United States. Can you name the other three? Highlight the text between the arrows for the answer: >> The Cottonmouth (Water Mocassin), Rattle Snake, Coral Snake << As with most snakes, it has TRGHV symmetry. The black rat snake is a non-poisonous snake, and like the copperhead (and most other snakes with patterns), it has TRGHV symmetry.

 In the process of working with strip patterns, we have discovered that in fact, there is an eighth symmetry - the delightful MRT symmetry. Ha ha ha! No really ... I pity the fool who don't tesselate the plane ...

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 Wallpaper patterns are patterns of symmetry that tessellate the plane from a given fundamental region. There are seventeen different types of wallpaper patterns. In the examples below, you will see the fundamental regions highlighted, as well as the translation vector generators that can be used to complete the pattern by translation, after the other isometries of the pattern are completed.

 The Giant's Causeway, located in Ireland, is an fascinating *632 formation found in nature. It is a collection of hexagons tesselating the ground - even in 3D at some points. Bees form their honeycombs in a *632 pattern as well. There seems to be a lot of hexagonal symmetry in nature. Any conjectures on why that's the case? The answer lies with steiner points and minimal networks.

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 Bilateral symmetry is symmetry across a line of reflection. Are people symmetric? We think we are, but upon closer analysis, we are less symmetric than we think. The more simple the creature (ants --> elephants), the more likeley it is that it will be perfectly symmetric. We took two professors, cut and pasted half of their head in Photoshop, and flipped that half horizontally. We then aligned the two halves so that it came closest ro resembling a human head. You be the judge on how good of a job we did and how symmetric people around us are in general ...

 Here is our professor - Dr. Clint McCrory, who as you will see, is very symmetrical. Which side of Dr. McCrory do you think this is ... ? ... and this one? How quickly did you pick up on the differences?

 Dr. Larry Hatfield has a part on the side of his head which makes it easier to notice the symmetry involved. Here we can see his right side of the face reflected over the middle ... ... and the left side. As you can see, one is more predominate than the other. What are some things that would contribute to this asymetry?

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