Here I just want to point out that we can classify polygons drawn in the plane, not only by their number of corners, but by whether or not the polygon intersects itself or has notches. We can think of a polygon as a collections of points joined by sticks with no membrane filling in the result or we can include the interior of the polygon along with the 'sticks.' There are pros and cons for defining shapes in particular ways. When we classify polygons that are drawn on a flat piece of paper in the plane, we can do so by counting the number of corners of the polygon or by counting the number of sides (edges) of the polygon. A triangle is a collection of three points not on a line and the segments joining pairs of the points which are known as the vertices of the triangle. Here we will give special consideration to the notion of a polygon.Īfter the point and the line, among the most fundamental of geometrical objects is the triangle. In describing a shirt one might use terms like sleeves and buttons and in describing a car one might mention windows and wheels. Sometimes these parts can be viewed with terms that overlap. These parts might be described as: points, lines, membrane patterns, corners, curves, etc. One might try to understand geometric objects in terms of the parts that make them up. The eye responds to issues related to the length of segments and the area of regions, even if sometimes the fact that area scales as the square of length rather than length itself sometimes causes individuals to make misleading judgments about diagrams. Proofs of theorems using 'physical models' such as the diagrams in Figures 1 and 2 are quite compelling because of the amazing ability of humans to input and process visual information. Here are diagrams (Figures 1 and 2) that support one of the many proofs of the Pythagorean Theorem that involved moving around pieces of squares and assembling them to form other squares.įigure 2 (A proof of the Pythagorean Theorem based on reassembling the pieces of squares on the sides of a right triangle, shown in white. But this theorem about lengths can also be interpreted as a statement about the areas of the geometric squares that can be constructed on the sides of a right triangle. The Pythagorean Theorem states that if one has a right triangle (one where two sides meet at a 90 degree angle-that is, are perpendicular), the square (in the algebraic) sense of the lengths of the side opposite the right angle is the sum of the squares of the lengths of the other two sides. Geometric Decompositionīefore addressing the issue of geometric compositions in earnest, as a teaser remember that one of the most important and well known theorem in mathematics is the Pythagorean Theorem, though attributing it to Pythagorus or even the Pythagoreans distorts the history of this remarkable result, which can be viewed as a result in algebra or a result in geometry. This suggests something that cannot be broken up into parts. For example, consider the word 'irreducible'. It is not uncommon in mathematics to use words as technical vocabulary that suggest ideas that a word has in more ordinary usage, that is non-mathematical contexts. Words connoting or related to decomposition in English are: decompositions, dissections, factoring, irreducible, etc. There I also looked at partitions of positive integers-for example, $5 = 4 + 1$ and $5 = 3 + 1 + 1$ are but two of the partitions of 5.

In an earlier Feature Column essay I looked at how by studying primes 2, 3, 5, 7, … we get insight into big integers such as 1111113.

Perhaps these people might decide it was a small moveable house? If in some future state of the Earth there were no automobiles and some humans came across a well preserved car from the 1970’s, but with no prior knowledge of what such a thing was, how might they interpret what they were looking at? An archaeologist at that time might try to understand its parts as a way to think through what the whole thing was good for. When looking at a body of mathematical ideas, one might look for the “atoms” or parts so that one could see the whole by having insight into its parts.

A remarkable theorem involving decompositions is that if one has two plane simple polygons of the same area, it is possible to decompose either of the polygons into polygonal pieces that can be reassembled to form the other polygon… Geometric Decompositions

Explain decompose and decompose math