Proof That Only 3 Regular Polygons Tessellate

Tessellation is the tiling a surface (in this case a flat surface) without gaps and overlaps. This concept is very important since many of the floors and walls are tiled nowadays. But tessellation is a lot more than that. It actually obeys the law of mathematics particularly angles.

The plane can be tiled with just one polygon as shown below. Some of such types of polygons are squares, triangles, and hexagons.

Can you think of other shapes that can tile the plane individually?

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Proof that -a = -1(a)

In the elementary school and high school mathematics, we are usually treating some statements as facts (or axioms). In reality, many of these statements are actually theorems. This means that if you want to go deeper, it is good to read the proofs of these theorems.

Note that all theorems in mathematics, even the most obvious, have their mathematical proofs just as the one we will prove below.

In this proof, we show that the the product of any number and -1 is equal to the negative of that number. That is, if a is a real number then -a = -1(a). Please refer to The Axioms of Real Numbers to understand more about the explanation below including the numbered axioms.


-a = -1(a)


Notice in the proof that on the left hand side, we have (-a). We manipulate the right hand side to become -1(a).  Continue reading…


Parallel Lines and Ratio Proof

Parallel lines have many important properties that you have to memorize in order to be successful in proving problems. In this post, we will discuss the properties of parallel lines intersecting two other lines. We are going to show the relationship among three parallel lines intersecting two lines.

In the figure below, parallel lines l, m, and n, intersect line a at points A, B, and C, and intersect line b at points D, E, and F. The theorem is that the ratio of AB to BC is the same as the ratio of DE to EFContinue reading…

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