How Pi Can Save Your Life: Using Math to Survive Plane Crashes, Zombie Attacks, Alien Encounters, and Other Improbable Real-World Situations

By Chris Waring

Math to the rescue! Discover equations for all sorts of real-world situations in this fun guide to surviving an unpredictable world.
 
Learn how to survive absurd yet statistically possible scenarios using the all-encompassing power of mathematical equations!
 
Whether you paid much attention in math class or not, the inescapable truth is that life is full of equations. You use differentiation when driving from point A to B and apply basic geometry when you’re crossing the road between traffic, even if you don’t realize it. But what if you were plummeting to your death in a plane with no engine and needed to know what size parachute to make from your cabinmate’s sari in order to survive? Math teacher Chris Waring tackles some frankly ridiculous scenarios with essential, bulletproof equations so you can learn to:
 

• Communicate with an alien civilization
• Save your town from a zombie apocalypse
• Contain a major oil spill
• Excavate a fossil that could be a major scientific discovery
• Pilot a space shuttle back to earth
• Guard a priceless painting in the Louvre
• Perform Hollywood stunts in a blockbuster action film, and more!

 
Praise for Chris Waring
 
“Exploring the mysteries of math has never been so much fun.” —Lancashire Evening Post

• Language: English
• Publisher: Open Road Integrated Media
• Release date: Aug 31, 2021
• ISBN: 9781646042234

Chris Waring

Chris Waring studied for a degree in Mechanical Engineering at Imperial College, London before becoming a maths teacher. Since then he has taught small children and Oxbridge candidates and everybody in between. He is the author of An Equation for Every Occasion, Maths in Bite-sized Chunks and From Zero to Infinity in 26 Centuries, also published by Michael O'Mara Books.

Book preview

CHAPTER 1

Born to Louvre

Your reputation as a private security consultant is second to none, and some of your more high-profile cases have recently caught the attention of the international media. When a woman dressed in the height of Paris chic arrives at your office, you quickly clear your schedule. You offer her a drink and agree to help her determine which member of her museum staff is stealing masterpieces and replacing them with nearly identical forgeries. The Louvre has a tight budget in terms of security, and together you must work out how to monitor the newly opened Salle d’Art Mathématique exhibition with as few guards as possible. She requires every part of the room to be within the line of sight of at least one guard at all times to keep the various paintings, sculptures and other artworks safe. How can you do this in the most efficient way possible?

This will require a bit of mathematical logic and geometrical reasoning. If you start to think about the room and the guards in mathematical language, you can represent the number of guards needed as g and then see whether you can narrow down the value of g at all. To do this, you’re going to need to consider polygons.

Polygons are two-dimensional shapes with straight sides. The floor plan of most rooms comprises polygons and traditionally involves mostly right angles, though this isn’t always the case, as we can see with the floor plan of the gallery.

Polygons are described according to the number of sides they have. A triangle is a polygon with three sides, which happens to be the lowest number of sides possible for a polygon. If you stick two triangles together, edge to edge, you can make a four-sided shape known as a quadrilateral. Quadrilaterals include rectangles, squares, trapezoids, kites, parallelograms and rhombuses. Add another triangle and you have a pentagon, a five-sided polygon. If you keep adding triangles, you add additional sides to your polygon.

Polygons can be convex or concave. Convex polygons are ones where the interior angles at the corners of the shape are all less than 180°. This means that, if you are looking at the polygon from outside, its edges would appear to extend outwards towards you, which is what convex means. Essentially, the pointy bits would stick out. With concave polygons, at least one of the interior angles is more than 180°, which means there would be bits that, from the outside, appear to point inwards.

Imagine yourself standing inside a room with a convex polygon floor plan. You would be able to see everywhere in that room, no matter where you stood. Mathematically, if you stand at any point in the room, you could draw a straight line from yourself to anywhere else in the room. In this context, the line represents your eyesight. Therefore, one guard can supervise any convex room.

Sadly, perhaps in an attempt to be artistic or maybe just to increase hanging space, the Salle d’Art Mathématique is a concave polygon with twenty-eight sides—an icosioctagon, to be precise. There is no point inside it where you can draw a straight line to every part of the polygon, so we can definitely say that more than one guard is required to watch the entire room. So, we now know that g > 1. Perhaps a little obvious, but we’ve got our starting point.

We know that you can build polygons out of triangles. As you might also recall from school, the angles within a triangle always add up to 180°. As triangles have three angles, each angle must be less than 180°, which means that triangles cannot be concave. Therefore, no matter the triangle, a single guard can always see the whole of a triangular room. This is not true for quadrilaterals or polygons with more than three sides, of course, as they could be concave. So, you now know that your client will require, at most, one guard for every triangle that makes up the polygon. It might be useful at this point to mention that a polygon is always made up of two fewer triangles than it has sides: a triangle has three sides and (obviously) one triangle, a quadrilateral has four sides and contains two triangles, a pentagon has five sides and three triangles, and so on.

So, for a room with n sides, g would need to be at least n − 2, which gives me g ≤ n − 2. If you combine this with our previous statement for g, you will see that

1 < g ≤ n − 2

For the gallery predicament, where n = 28, you can see that this becomes

1 < g ≤ 26

So, one way of splitting the room into triangles would look like this:

There are other ways, of course, but you can be sure that a 28-sided polygon will have 26 interior triangles.

The client seems happy with your logic, but has some understandable concerns about having twenty-six guards loitering in her gallery, even if her budget could stretch to it. But you can reassure her that your work is not yet finished, and with a bit more effort you can bring this number down considerably.

Let’s think about what would happen if you placed the guards in a corner of each of the triangles. If we label the corners of each triangle A, B or C, you will find it is possible to do it in a way that no two corners with the same letter are adjacent to each other.