I was drawn to this chapter in our textbook in part because of an interest I have in 3D computer modeling. Polygon modeling is one way of creating 3D objects with software such as Maya and 3D Studio Max. The creation and manipulation of polygon vertices, edges and faces through rotation, positioning and scaling is one way of building characters and objects for movies and games. This paper starts with definitions of basic terminology related to the subject of polygons, tiling with the pentagon in particular.

A ** polygon** is a shape that can be drawn on a

**(**

*plane**a 2 dimensional surface*) so that the starting and ending points are the same and the path never crosses or retraces itself. We will call the line segments that compose a polygon

**, and the end points of the sides,**

*sides***. Polygons include the triangle(3 sides), Quadrilateral (4 sides),Pentagon (5 sides),and Hexagon (six sides), there are many more and as the number of sides increase they are often refered to simply as**

*vertices***.**

*N-gons*If you can cover an entire *plane* with a polygon shape so that no gaps exist, it is said to be *tileable**. *A** Regular Polygon **has sides that are all of equal length, and the angles at each vertex are all of the same degree. The square and the equilateral triangle are 2 examples of

**. An**

*Regular Polygons***side lengths and vertex angles that are not all identical.**

*irregular Polygon has*An ** edge to edge tiling** is one in which polygons tile a plane without gaps and the entire length of a polgon’s edge is matched by the edge of another polygon. A good example of a tileable pattern that is not

*edge to edge*is a brick wall, the edge of one rectangular polygon is staggered and a corner vertex of one polygon rests midway along the edge of another identical rectangular polygon.

The equilateral triangle, square and hexagon all work to form *edge to edge* *tiling,* this is because the sum of the angle degrees of each of these types of regular polygons will form 360 degrees. One equilateral triangle is composed of three interior angles (vertex angles) each of which is 60 degrees. Six of these triangles can be placed so that one vertex of each triangle meets at a central point.

A square composed of 4 90⁰ interior angles can be combined with three other identical squares, all sharing a common vertex, the sum total of all 4 angles equaling 360 degrees. A regular hexagon has six identical interior angles each of which is 120 degrees. Three of these hexagons can be placed together so that one vertex of each hexagon is shared by the other two. 120⁰ x 3 equals 360⁰ and once again no gap exists around the vertex.

A regular pentagon has five interior angles (vertex angles) all of which are 108⁰. If three regular pentagons of the same size are placed together so that one vertex is shared by all three, a gap between two of the sides will exist because the sum of the interior angles radiating from the shared vertex is only equal to 324⁰ rather than 360⁰. We are unable to squeeze 4 regular pentagons sharing a vertex together because the sum total of all the interior angles would equal 432⁰, and the total degrees radiating from a single vertex is only 360⁰.

So we have learned a little about tiling a few *regular polygons. What about tiling with irregular polygons* (polygons where all side lengths and interior angles are not equal)? The interior angles of any triangle always equal 180⁰. If all 3 interior angles are brought together at one vertex the combined sum will be 180, this factors into 360, so we find that any irregular triangle is tileable. This is also true for any quadrilateral because the sum of all 4 interior angles always equals 360⁰.

We discovered earlier that a regular pentagon is not tileable, but there have been 14 irregular pentagons found to date that are tileable. It is conceivable there are more that have yet to be discovered.

Karl Reinhardt (1895-1941) discovered five pentagon tilings in 1918. R.B. Kershner found 3 tilings in 1968. Richard E. James found 1 in 1975. Marjorie Rice an amateur mathematician found 4 between 1976 and 1977. The 14^{th} was found by Rolf Stein in 1985. These are the known convex pentagonal tilings. It is also possible to have concave pentagonal tilings.

Some of the convex pentagon tilings have been given names, this includes the Cairo pentagonal tiling. This Isohedral (all faces are the same) tiling was given this name because some streets in Cairo are paved with the design. The interior angles of this type of pentagon are 120⁰, 120⁰, 90⁰, 120⁰, 90⁰.

The Floret pentagonal tiling is given its name because it has six pentagonal tiles that radiate outwards from a central point, much like the petals of a flower. John Horton Conway author of an early computer game named “the Game of Life” calls this the 6-fold pentille. Each face has four 120⁰ interior angles and one 160 ⁰ angle.

The Prismatic pentagonal tiling has three 120⁰ angles and two 90⁰ angles

Here are a few more convex irregular polygons with the interior degree angles labeled that have been found to be tileable.

This particular tiling of congruent pentagons was discovered by Marjorie Rice in 1995. Mathematician Doris Schattschneider adapted the tiling for use in a building’s foyer.

The picture above shows the 14 known irregular pentagons that are tileable

Left –Natural Geological Formation,The Giant’s Causeway, in Northern Ireland

Center-Pompeii Tiling

Right-Archway from Darb-i Imam Shrine, Iran. Built 1453 C.E.

Tiling of polygons in intricate ways is ancient. It is exciting that new patterns have recently been found and that new ways of conveying the underlying concepts behind them are being formed. I have only begun to scratch the surface in an exploration of polygons and tilings. If you found this topic interesting may I suggest you take a look at some of the sources this information was gathered from.

**A Mathematical View of Our World ** – textbook for the class *Explorations in Mathematics*

http://demonstrations.wolfram.com/PentagonTilings/ -This is an interactive site. To use it you need to download the interactive player. What is most interesting is that it conveys very well that for some of the irregular pentagon tilings the interior degree angles are not fixed. Sort of like the gradual collapsing of a cardboard box.The site does a nice job with sliders to convey this idea.

http://facultypages.morris.umn.edu/~mcquarrb/teachingarchive/M1001/Resources/Lecture20a.pdf

http://mathtourist.blogspot.com/2010/06/tiling-with-pentagons.html

http://www.scipress.org/journals/forma/pdf/1501/15010075.pdf

http://www.oocities.org/liviozuc/pentagons.html

http://en.wikipedia.org/wiki/Polygon

There are other types of tilings. I would highly recommend taking a look at the two videos below to widen ones horizens related to tessellation and Penrose tilings.

http://www.youtube.com/watch?v=xx8tLKAs87I&feature=related Oragami Tesselation

http://www.youtube.com/watch?v=uxlTvWoFf20&feature=relmfu 3D Computer Animated Penrose tiling

http://www.sciencenews.org/view/generic/id/8270/title/Math_Trek__Ancient_Islamic_Penrose_Tiles

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