Fundamental Concepts

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Periodic table of the elements

Relevant examples from Escher's work:

Explorations

Begin learning about geometry with:

Classification

Throughout science, one of the fundamental problems is that of classification. For example, chemists classify chemical compounds and reactions and biologists classify species. The problem of classification requires discovery and organization. Biologists discover new species all the time, and usually these fit into the traditional organization of kingdom, phylum, class, order, family, genus, species. Occassionally, new discoveries suggest changes to the organization. For example, the number of biological kingdoms has changed many times and is still a matter of some debate (wikipedia:Kingdom (biology)).

Astronomers are interested in classifying stars. They discover stars with telescopes and other equipment, then group the stars into classes. These classes have both descriptive names (white dwarfs, dwarfs, giants, etc.) as well as numbers (Ia, Ib, II, III, etc.) The picture below shows a plot of 22,000 stars by temperature (on the x-axis) and luminosity (on the y-axis). When organized in this way, the logic behind the different classes is easy to see.

Classification of stars

In mathematics, classification problems also require discovering examples and finding organization. Mathematical classifications have one additional feature - completeness. It seems unlikely that biologists will ever discover all living species or that astronomers will ever identify every star (or know if they have accomplished these feats). However, a good mathematical classification comes with an argument, or proof, that there is nothing left to discover.

Escher spent a great deal of effort on classification problems. He was interested in classifying symmetries, tessellations, polyhedra, stamping patterns, linked circles, and probably many more mathematical collections. One clear reason for his interest was that he used these concepts in his artwork and wanted to discover all possible variations. But we will see that Escher must have also been interested classification for the same reasons as scientists - knowledge and understanding of the world. Later in the course will follow his lead in studying symmetries, tessellations, and other concepts. But first, for practice, we will look at some simple classification problems involving polygons.

Triangles

Most of the mathematics in Escher's work is related to geometry. One of the fundamental objects in geometry is the triangle, and in this section we will explore various ways to classify triangles.

First, we introduce two notions of "sameness" for triangles:

Congruent triangles
Two triangles are congruent if they have the same side lengths and angle measures. They are the same size and shape.
Similar triangles
Two triangles are similar if they have the same angle measures. They are the same shape, but may be different sizes.

One simple classification comes from considering angles:

Triangles.svg
Acute triangle
An acute triangle is one in which all the angles are acute (less than 90°).
Obtuse triangle
An obtuse triangle is one in which one of the angles is obtuse (more than 90°).
Right triangle
A right triangle is one in which one of the angles is a right angle (exactly 90°).

It is not hard to see that every triangle falls into exactly one of these three classes. Every triangle is either acute, obtuse, or right.

Considering side lengths leads to other classes of triangle:

Equilateral triangle
An equilateral triangle is one in which all three sides have the same length.
Isoceles triangle
An isoceles triangle is one in which two sides have the same length.
Scalene triangle
A scalene triangle is one in which all three sides have different lengths.

This classification is of a different nature than the angle classification because equilateral triangles are also considered to be isoceles. That is, the class of equilateral triangles is contained in the class of isoceles triangles. In this classification, not every triangle falls into exactly one of the classes.

As a side note, a triangle with two equal angles must be isoceles, and a triangle with three equal angles (equiangular) must be equilateral.

Polygons

Kasimir Malevich, Suprematism, 1915. Oil on Canvas. Dimensions: h 101.5 w 62 cm. Now in the Stedelijk Museum in the Netherlands.
Polygon
A polygon in the plane is a closed figure made by joining line segments. The segments may not cross, and each segment must connect to exactly one other segment at each endpoint.
Polygons
Polygons.svg
Not Polygons
Notpolygons.svg

The left figure is not closed, and the figures in the middle are not made of line segments. The figure on the right is not a polygon, since its sides intersect each other.

Vertex
A vertex of a polygon is a point where two sides come together.
Vertices.svg

The word "vertex" is more precise than the common term "corner", because "corner" has many other uses in English. The plural of "vertex" is "vertices" - a triangle has three vertices.

A fundamental characteristic of any polygon is the number of sides it possesses, which is the same as its number of angles. Classifying polygons by number of sides is important enough that there are special words for polygons with small numbers of sides:

# of sides Name
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Enneagon
10 Decagon
A 28-gon

There are (ridiculous) names for polygons with many more sides (see wikipedia:Polygon), but generally for larger numbers of sides, one uses the number of sides followed by "-gon".

One very special type of polygon is a regular polygon:

Regular polygon
A polygon with all sides having equal length, and all angles having equal measure.

Quadrilaterals

After triangles, the type of polygon we will encounter the most is the quadrilateral:

Quadrilateral 
A polygon with four sides.

There are names for many special classes of quadrilaterals:

Rectangle
A quadrilateral having four right angles.
Square
A quadrilateral with four equal-length sides and four right angles.
Parallelogram
A quadrilateral with two pairs of parallel sides.
Rhombus
A quadrilateral having all four sides of equal length.
Trapezoid
A quadrilateral having exactly one pair of parallel sides. The two sides that are parallel are called the bases of the trapezoid.

In the Quadrilaterals Exploration you learn the relationship between these (and other) special classes of quadrilateral.

In the Triangles and Quadrilaterals Exploration you will build quadrilaterals out of triangles, and relate special triangles to special quadrilaterals.

Convexity

A non-convex region. The line joining the two red points leaves the region.

A region in the plane is called convex if any two points in the region can be joined by a straight line segment that stays in the region. A region which is not convex is called non-convex rather than the colloquial "concave", which is slightly misleading when referring to the precise mathematical definition. As a general rule of thumb, non-convex regions have dents in them.

Here are examples of convex and non-convex regions.

Convex regions Non-convex regions
Regions-convex.svg Regions-nonconvex.svg

Sums of reciprocals

Asterisk.svg This section is optional.

This section presents a classification problem that simply answers a question about numbers. It is interesting here because the answer is straightforward but not obvious, and also because we will find a surprising connection to this problem much later in the course.

Question: In what ways is it possible to represent 1/2 as a sum of two reciprocals?

Let's make this question a little more precise:

For which whole numbers m and n is 1/m+1/n=1/2?

The process of discovery starts with some guesses. For example, if m=1 and n=3 then 1/1+1/3=4/3, which is too big. If m=4 and n=6 then 1/4+1/6=5/12, which is too small.

At this point, you can probably work a little bit and find that

{\frac  {1}{4}}+{\frac  {1}{4}}={\frac  {1}{2}} and {\frac  {1}{3}}+{\frac  {1}{6}}={\frac  {1}{2}}.

The final step is to prove that these are the only two ways to get 1/2 in this manner. Here is an argument.

We'll use m for the first number and n for the second. First consider letting m be either 1 or 2: Since

{\frac  {1}{1}}+{\frac  {1}{n}}>{\frac  {1}{2}} and {\frac  {1}{2}}+{\frac  {1}{n}}>{\frac  {1}{2}},

we see those values won't work for m; and, similarly, they won't work for n, either. Thus, we only need to look at numbers m and n both larger than 2; that is to say, we can assume m\geq 3 and n\geq 3.

From m\geq 3, we derive {\frac  {1}{m}}\leq {\frac  {1}{3}}. In other words, 1/3 is the biggest our first reciprocal can be. What does that leave for our second reciprocal, given that they must add up to 1/2? Since

{\frac  {1}{2}}-{\frac  {1}{3}}={\frac  {1}{6}},

we see the second reciprocal must be at least 1/6. This means that any choice of n>6 will leave the sum too small: We conclude n\leq 6. Similarly, we must also have m\leq 6.

This greatly simplifies our search, because now we only need to look at values of m and n from 3 to 6. We can check all of these values by hand. The table below summarizes the results, with red squares indicating sums larger than 1/2 and blue squares indicating sums smaller than 1/2.

m\backslash n 3 4 5 6 7
3 2/3 7/12 8/15 1/2 10/21
4 7/12 1/2 9/20 5/12 11/28
5 8/15 9/20 2/5 11/30 12/35
6 1/2 5/12 11/30 1/3 13/42
7 10/21 11/28 12/35 13/42 2/7

This finishes the classification:

The only solutions are 1/4+1/4=1/2 and 1/3+1/6=1/2.

The table is full of other patterns. To get into the spirit of discovery, spend some time looking for them. Which ones continue if the table is extended past 7? What about if rows and columns for 1 and 2 are added?

For more interesting questions and a little history lesson, refer to wikipedia:Egyptian fraction.

Exercises

Polygon Exercises