Following are some items relating to geometry discussed in the history of mathematics.

 Art and Geometry The Mobius Strip Euler's Formula Tangrams Tessellation Polyhedron

Art and Geometry

What is art? Well, everyone asked this question would have a different answer, because we all have different likes and dislikes. Each and every culture in the world evaluates art and how it relates aesthetically to their surroundings and/or beliefs. Aesthetic understanding of an artwork is the combination of the ability to see, interpret, and evaluate it. Therefore, one person might have a different viewpoint of an artwork than someone from another culture.

In history, the Greeks were believed to be the supreme culture. However, William M. Ivins, Jr. studied the art of the Greeks and also their geometry. In his book, "Art and Geometry: A Study in Spatial Intuitions," Ivins creates a controversial study to the above myth. According to Ivins, the Greeks were "tactile minded," meaning that they created works of art that were perceived through the sense of touch. The Greeks "tactile" world view is visible in their art by the lack of motion, emotional and spiritual qualities.

Ivins goes on to say that the Greeks form of art was the result of not completely understanding the laws of perspective. So, what is meant by "the laws of perspective?" Well, to put it simply, it means the proper technique for representing a three-dimensional object on a two-dimensional surface.

Artists of the Renaissance period were the first to be successful in perspective. In 1636, a man named Girard Desargues introduced his "perspective ladder." This was used by artists as a tool for bringing perspective to their work.

Just as the Greeks based their art on tactile qualities, they didn't stray far from this way of thought in their geometry. They believed that parallel lines stay parallel forever. So their lack of modern thought for geometrical continuity and perspective left the Greeks at a disadvantage in the mathematical field. Geometry progressed through time to involve perspective geometry. Following is an example of a line-divider. This helps bring perspective to line designs and optical art, which utilizes geometry.

Today, artists often use geometrical elements such as lines, angles, and shapes to create a theme throughout their artwork. Also, artists started using these geometrical elements as a way to create the illusion of the third dimension. This art is known as Optical or Op Art. The following is an example of optical art.

Students should start out their study of optical art, by creating line designs and working with symmetry. See example of line designs below. Then students can apply the concept of shading to their designs to create a sense of perspective. Students will be building their spatial intelligence for understanding advanced mathematics.

Contributed by Lanetta J. Burdette

Reference:

Ivins, Jr., W.M. "Art and Geometry: A Study In Spatial Intuitions" (1946) Dover Publications, Inc. New York: NY.

Seymour, D., Silvey, L. and Snider, J. "Line Designs" (1994). Ideal School Supply Company. Alsip, IL.

Thompson, K. and Loftus, D. "Art Connections: Integrating Art Throughout the Curriculum" (1995) Good Year Books. Glenview, IL.

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The Mobius Strip

Biographical Information

August Ferdinand Mobius was born 1790 in Saxony (now Germany), and died in 1868, in Leipzig. His father died when he was three years old. Mobius was educated at home by his mother until he was thirteen, when he went to college in Saxony. He graduated from the college in 1809, and became a student at the University of Leipzig. His mother wanted him to become a lawyer, but he chose to study math, astronomy and physics instead. Mobius learned from only the best teachers. In 1813 Mobius studied under Gauss, the director of the observatory in Gottingen. He then continued his studies but under Johann Pfaff, who also taught Gauss. The year of 1816 brought an appointment to the Chair of Astronomy and Higher Mechanics at the University of Leipzig. The University granted Mobius a full Professorship in Astronomy in 1844. He stayed at the University for the remainder of his career.

The Mobius Strip

Mobius was a pioneer in the field of topology. Topology is the study of “those properties of geometric figures that remain unchanged even when under distortion, so long as no surfaces are torn.” It defined a property of simple closed polyhedra pertaining to the vertices (V), edges (E), and faces (F): V - E + F = 2.

Mobius speculated that a polyhedron was a collection of joined polygons. This speculation introduced the notion of “2-complexes.” It was this study that led Mobius to the surface now known as a Mobius Strip: the simplest geometric shape, a one sided surface. Mobius is best known for this development. It may be replicated by taking a strip of paper or ribbon, turning one side 180 degrees long ways and attaching the two ends. The paradox of the Mobius Strip is that a one-surfaced, one- edged figure is three dimensional. This very paradox, with derivations such as the Klein Bottle, may be used to define such celestial anomalies as black holes and worm holes.

To view several different examples the Mobius Strip, refer to the web sight credited in the reference section.

Contributed by Steve Bixler

References:

1. Historical Topics for the Mathematics Classroom. Thirty-First Yearbook, Washington D.C., NCTM, 1969.
2. Boyer, Carl B. A History of Mathematics (2nd Edition). John Wilcox and Sans Inc. 1968, 1989, 1991, New York.
3. http://www.mhri.edu/~pdb/geometry/mobius/

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Euler's Formula

In the mid-nineteenth Century, a new development in Geometry called Topology started taking shape (no pun intended!). Topology is the study of geometric figures that persist even when the figures are subjected to change in such a way that their properties are lost. A few isolated discoveries before the mid-nineteenth century have become known throughout the modern development of Topology. One of the most important is a formula showing the relationship between vertices, edges and faces in simple polyhedra. The generalizations, which have become known as "Euler's Formula, takes it's place among the central theorms of geometry.

Euler's formula is one of the most important theorems of Geometry, with fifteen different proofs surfacing since it's inception, first discovered by Descartes and later rediscovered by Euler, to whom we credit the theorem, in 1752. Showing a relationship between vertices, faces and edges of simple polyhedra, Euler interest was to classify polyhedra. Euler presents his theorem as the number of vertices, plus the number of faces, minus the number of edges of any simple polyhedra will equal two; V + F - E = 2. From his findings, it has been determined that there exists only five platonic solids which can be constructed by choosing a regular polygon and having the same number of shapes meet at each corner.

The Five Platonic Solids include:
 1. Tetrahedron, which has three equilateral triangles at each corner. 2. Cube, which has three squares at each corner. 3. Octahedron, which has four equilateral triangles at each corner. 4. Dodechaderon, which has three regular pentagons at each corner. 5. Icosahedron, which has five equilateral triangles at each corner. The figures above are reproduced with the permission of George Hart

Can you satisfy Euler's Formula for the above geometric shapes? The cube, for instance, has eight vertices, six faces, and twelve edges or 8 + 6 - 12 = 2. All five of these shapes can be found in nature. The cube, tetrahedron, and octahedron can be found in crystals while the dodecahedron and icosahedron can be found in certain viruses and radioilaria. This would be a wonderful way to integrate math with science. For more information including the fifteen proofs of Euler's Formula, visit the following web sites:
http://www.ics.uci.edu/~eppstein/junkyard/euler/
http://www.cut-the-knot.com/do_you_know/
http://www.georgehart.com/virtual-polyhedra/platonic-info.html

Contributed by Jan Swanson

References:

1. Bogomolny,A. (2000). Regular polyhedra. Retrieved June 12, 2000 from the world wide web: http://www.cut-the-knot.com/do_you_know/.
2. Bunt, L., P.S. Jones, J.D. Bedient. (1976). The historical roots of elementary mathematics. New York: Dover Publications, Inc.
3. Dunham, W. (1990). Journey through genius. New York: John Wiley & Sons, Inc.
4. Eppstein (May, 2000). Fifteen proofs of Euler's Formula. Retrieved June 12, 2000 from the world wide web: http://www.ics.udi.edu/~eppstein/junkyard/euler/.
5. Hart, G. (2000). The five platonic solids. Retrieved June 12, 2000 from the world wide web: http://www.georgehart.com/virtual-polyhedra/platonic-info.html
6. Kline,M. (1972). Mathematical thought. From ancient to modern times, volume 3. New York: Oxford University Press.

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Tangrams

What is a Tangram?
Tangram is an ancient, unique, Chinese puzzle that consists of seven (geometric) pieces: one square, five triangles and one parallelogram. Of the five triangles there are two large, two small and one medium in size. The large triangle is twice the area of the medium triangle. The medium triangle, the square, and the parallelogram are each twice the area of a small triangle. Each measure of the square measures 90°. Since each triangle contains a 90° and two 45° angles, they are isosceles right triangles, and the two sides opposite the 45° angles are congruent. The parallelogram contains 45° and 135° angles. The relationship among the pieces enables them to fit together to form many figures and arrangements.

Here is a pattern for making the pieces for a tangram.

What is the history behind the Tangram?
The exact origin of the Tangram is unknown. It is more than 4,000 years old. However, there are many interesting stories as to how it came about. One such story claims that a large pane of glass was ordered by a king. When the large, perfect square glass frame was in transit to the kings castle it was dropped and surprisingly it had not shattered into thousands of pieces, it had broken into seven perfect, geometric shapes. When they tried to reassemble the seven pieces they found they could make many other designs. They proceeded to the castle and presented the broken glass as a puzzle for the king. The king was fascinated with the glass puzzle. The invention of the Tangram puzzle is not actually known. The earliest mention of it was found in a book dated in 1813 A.D. At this time the puzzle was already considered to be "old". This puzzle was originally considered to be a game for women and children. This would have made it unworthy of being studied or written about. The puzzle was introduced to the West in the mid 1900th century to sailors who were involved in trade with China. They too, were intrigued by the simple yet intricate puzzle.

Tangrams today . . .
Tangrams are still entertaining and frustrating today. This puzzle continues to attract people of varying intellectual levels. Those interested in math enjoy it for its geometry and ratios. Most children are attracted by the how simplistic the pieces are and that there are no set solutions, it is a free form activity. This classic puzzle still attract players, both young and old. Construct a tangram puzzle of your own by using a 4 x 4 inch grid. (Make the cut lines to resemble the angles and lines of the diagrams at the top of this page.)

Rules of the puzzle:
 Classic rules state that all seven pieces must be used. All pieces must lie flat. All pieces must touch. No pieces may overlap. Pieces may be rotated and/or flipped to form the desired shape.

Here are some puzzles to try:

References:

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Tessellation

Have you ever wanted to create a work of art but did not know where to begin? Think about what types of relationships exist between art and mathematics. There are many ways in which artists use math. I will give you a few examples. Art that depicts height and width is a two-dimensional design. Three-dimensional art shows height, width, and depth. Art also uses proportions, patterns, and geometry. Proportion is the relationship of a part to the whole or another part. Patterning plays a big role in the developing in art. Next time you go to an art museum, look closely at the pieces and try to find the patterns and mathematical influence.

The repetition of a pattern is called a tessellation. M.C. Escher, a well-known artist, used the tessellation concept in many of his artworks. The tessellation concept consists of redrawing a shape using slide, reflection (flip), and rotation (turn). The point in which three or more tiles meet in a tessellation is called a vertex. Triangles, squares, and hexagons are regular polygons that tessellate by themselves. This can be proven mathematically. A full rotation is 360o. Using an equilateral triangle, with angles of 60o, 6(60o)=360o. This calculation proves that six tiles meet in the vertex of a tile tessellation. Four tiles meet in the vertex of a square; 4(90o)=360o. A hexagon with angles of 120 degrees has three tiles that meet in the vertex; 3(120o)=360o. Different types of regular polygons can be used to tessellate polygons such as the pentagon, heptagon, and octagon.

For a tessellation, choose one or two geometric shapes. Make a tessellation by sliding, reflecting, or rotating the shape. After creating a pattern, or picture, add color and texture to the shapes. The picture represents a work of art with a mathematical foundation. Can the pattern be altered by changing the appearance of the shape or shapes? Try to create a different look by using the same geometrical shape or shapes with a slight variation and see the difference in the end result. The exploration of tessellation can be very exciting.

Below is an illustration demonstrating a tessellation. The three simple shapes show the slide, reflection(flip), and rotation (turn) techniques. For explanation purposes, start with the middle tile. The slide technique is utilized to redraw the middle tile to the tile above it. From the top, the tile is reproduced clockwise using the reflect, slide, reflect and rotate techniques consecutively.

Contributed by CiCi Naifeh

References:

1. http://library.thinkquest.org/16661/escher.html
2. http://library.thinkquest.org/16661/escher/tessellations.1.html
3. Herberholz, David and Barbara. ARTWORKS for Elementary Teachers, Developing Artistic and Perceptual Awareness. The McGraw-Hill Companies, Inc., 1998.
4. Wells, David. The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, 1991.

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Polyhedron

A polyhedron is a geometric figure which is the three-dimensional version of the plane polygon. Another way to say it is that it is a finite connected set of polygons joined together in such a way that each side of every polygon coinsides (connects) with a side of exactly one other polygon.

The study of polyhedra was a popular study item in Greek geometry even before the time of Plato (427 - 347 B.C.E.) In 1640, Rene Descartes, a French philosopher, mathematician, and scientist, observed the following formula. In 1752, Leonhard Euler, a Swiss mathematician, rediscovered and used it.
 V - E + F = 2 V = the number of vertices, each point where three or more edges intersect. E = the number of edges, each intersection of the faces. F = the number of faces, each plane polygon.

This formula is true for simple polyhedra. A polyhedron is said to be simple if there are no holes in it; that is, the surface can be deformed continuously into the surface of a sphere. There are more complex ones that have their own formulas. In general, simple polyherons fall into two categories: convex and concave. A convex polyhedron is defined as follows: no line segment joining two of its points contains a point belonging to its exterior. A concave polyhedron, on the other hand, will have line segments that join two of its points with all but the two points lying in its exterior. Below is an example of a concave polyhedron.

The polyhedra that are most intriguing are the regular polyhedra. In a regular polyhedron all of the faces are regular polygons that are congruent. Furthermore, all the vertices of a regular polyhedron lie on the surface of a sphere. As it turns out, there are only five regular polyhedra and these are often referred to as platonic solids. The regular polyhedra are
 Tetrahedron 4 faces equilateral triangles Octahedron 8 faces equilateral triangles Hexahedron(Cube) 6 faces squares Icosahedron 20 faces equilateral triangles Dodecahedron 12 faces petagons

Contributed by Susan Eastman

References:
1. Courant, Richard and Robbins, Herbert, "What is Mathematics?", Oxford University Press, New York, 1996, p. 236
2. http://encarta.msn.com/find/Concise.asp?ti=06DC2000
3. "Polyhedron", Encyclopedia Americana, Grolier, Danbury, CT, V. 22, 1999
4. "Polyhedron", Collier's Encyclopedia, New York, V. 19, 1997

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