Proof that There Are Only Three Regular Tessellations

Tessellation is the process of covering the plane with shapes without gaps and overlaps. Tiles on walls and floors are the most common tessellations.

In this post, we are going to prove that the there are only three regular polygons that can tile the plane. Tiling the plane with regular polygons is called regular tessellation. These are equilateral triangles, squares, and regular hexagons. This proof only requires simple algebra. We will use the notation \{a,b \}, where a is the number angles of the polygon and b is the number of vertices that meets at a point. For example, a square has four angles, and at every point on the tessellation, four vertices meet at a point. So, we can represent square as \{ 4,4 \}. Another example is the regular hexagon. A hexagon has 6 sides, and at every point 3 vertices meet (see figure below), so a hexagon can be represented as \{ 6,3 \}Continue reading…


2015 Year in Review – 1st Semester

It’s July! Since it is half a year already, let us look back about what proofs we have learned so far.

For the past year, we have learned the proofs of 7 mathematical theorems, 3 in Geometry and 4 in Algebra. There are also three worked out proofs in triangle congruence. Congruence theorems like SAS, SSS, and ASA are used in these proofs.

Year in Review – 1st Semester

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Slope of Perpendicular Lines Theorem

If you have already learned about systems of linear equations, then you have probably discussed that the product of the slope of perpendicular lines is -1. The proof of this theorem comes from the fact that any point (x,y) rotated 90 degrees at about the origin becomes (-y,x). One example of this is shown below. The point (3, 4), when rotated 90 degrees counterclockwise becomes (-4,3).

With this fact, we prove this theorem.

Slope of Perpendicular Lines Theorem

If two lines with slopes m_1 and m_2 are perpendicular, then m_1 m_2 = -1Continue reading…