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# 1-Page Summary of The Unknown

## Overview

Kings, queens and wars are often mentioned in history books. However, equations aren’t usually included, although they have played a critical role in history. The Pythagorean theorem allowed us to draw accurate maps and Einstein’s theory of relativity changed the way we view time and space.

This passage will show you that although equations may seem abstract, they can reveal concrete relations in the world. The book talks about 17 of these key equations and here we’re going to talk about nine of them.

You’ll also learn that triangles, squares and circles are all the same shape. You’ll also learn how your car’s satellite navigation system can send you to a different planet. You will also discover how food processors can cause chaos. Finally, you will find out about an equation that caused a stock market crash.

## Big Idea #1: The Pythagorean theorem shaped our understanding of geography and Earth’s place in the solar system.

The Pythagorean theorem looks simple, but it’s actually more complicated than it appears. It’s made up of the equation a²+b²=c², which explains the relationship between two sides and the longest side in a right-angled triangle.

The Pythagorean theorem is a simple equation that allows us to calculate the length of any side of a right triangle. It’s vital because it leads to triangulation, which makes accurate measurements possible in mapmaking. Triangulation involves creating triangles and then measuring distances and angles between them.

Technological advancements have made it possible to measure things without triangulation. However, some methods still use the Pythagorean theorem. The Earth is a sphere, but what about the universe? It’s impossible to know for sure unless we can get outside of it. However, thanks to generalizations of the Pythagorean theorem and other mathematical formulas, we can find out more about our universe.

Pythagoras’s theorem is a triangle relationship in flat planes. However, it can be applied to curved spaces. For example, measuring triangles on the sphere makes it possible to measure our planet. In addition, those generalizations can tell us how curved the space is by measuring the triangles within that space.

## Big Idea #2: Euler’s formula revolutionized the world of math – and helps us understand DNA.

In geometry, a polyhedron is a three-dimensional solid consisting of faces joined together. One example would be the cube. Euler’s formula for polyhedra says that the number of vertices minus the number of edges plus two equals 2.

If we add up the numbers of faces, edges and vertices of a solid object then it holds true that F-E+V=2. This formula was discovered by Euler, who changed the way mathematicians think about geometry. His discovery led to the development of topology (a branch of mathematics).

In topology, shapes such as triangles, squares and circles are irrelevant. Lines can bend or stretch but they cannot be torn apart.

It’s a common question to ask if two geometric figures are topologically the same. This means that they can be deformed into each other by stretching and bending.

Sometimes, it’s hard to tell whether two solids are the same just by looking at them. To understand their topology, we can use a formula that tells us how many faces, edges and vertices they have. It’s called Euler’s Formula. If the number of faces plus edges plus vertices is 2, then they’re topologically equivalent; if not, they’re not.

This insight led to a field of mathematics that studies geometric objects. Its applications include biology, since the DNA in cells is shaped like a twisted ladder. Molecular biologists can use this math to understand how biological processes work and what shapes they take.

The Unknown Book Summary, by K.A. Applegat