Beautiful Mathematics

Martin Erickson
Series: Spectrum
Edition: 1
Pages: 192
https://www.jstor.org/stable/10.4169/j.ctt13x0n39

1. Front Matter
(pp. i-viii)
2. Preface
(pp. ix-x)
(pp. xi-xiv)
4. 1 Imaginative Words
(pp. 1-12)

The objects of mathematics can have fascinating names. Mathematical words describe numbers, shapes, and logical concepts. Some are ordinary words adapted for a specific purpose, such as cardinal, cube, group, face, field, ring, and tree. Others are unusual, like cosecant, holomorphism, octodecillion, polyhedron, and pseudoprime. Some sound peculiar—deleted comb space, harmonicmap, supremum norm, twisted sphere bundle, to name a few. Mathematical words have appeared in poems (see [19]). Let us look at some mathematical words.

Consider thelemniscate, a curve shaped like a figure-eight¹ as shown in Figure 1.1. We learn in [46] that it gets its name from...

5. 2 Intriguing Images
(pp. 13-24)

Many mathematical concepts are embodied in diagrams, drawings, and other kinds of images. A sketch may illustrate a theorem. A picture may point the way to new mathematics. Let us look at some mathematical images and learn about the mathematics behind them.

The equation

12+ 22+ 32+ … 242= 702

might seem, at first glance, to be a miscellaneous mathematical fact, but it is special. Édouard Lucas (1842–1891) posed a problem, called the Cannonball Puzzle, which asked for a numberNsuch thatNcannonballs (spheres) can be placed in a square array, or in...

6. 3 Captivating Formulas
(pp. 25-48)

Mathematical formulas, whether simple or complicated, convey in symbols the essence of mathematicians’ discoveries. Some formulas are well known, such as Euler’s formula e+ cosθ+isinθ. Some are less known. We will look at a few formulas I find beautiful, some stark and some ornate. You may find them beautiful too.

Here are three arithmetical curiosities:

123456789 × 8 + 9 = 987654321

123456789 × 9 + 10 = 1111111111

111111111 × 111111111 = 12345678987654321.

It is easy to verify their truth, but why do they work? What happens when you do the multiplication?

Heron’s formula,...

7. 4 Delightful Theorems
(pp. 49-82)

Mathematicians prove theorems. Once a theorem is proved, it is true for all time. The theorems proved by the ancient Greeks are as true today as they were over two thousand years ago, and the theorems proved today will be true even if, after millions of years, humans evolve into another species. In this chapter we present some delightful and sometimes surprising theorems.

Given any triangle, is it always possible to inscribe a square in it? We require that the square has a side on one of the sides of the triangle, with the other two corners touching the other...

8. 5 Pleasing Proofs
(pp. 83-108)

In mathematics, assertions can be proved, which distinguishes mathematics from other disciplines. Mathematical knowledge is thus absolute and universal, independent of space and time. In this chapter, we present some proofs that are particularly memorable. Most are not well known and deserve to be better known.

The Pythagorean theorem states that given a right triangle, the area of a square formed on the hypotenuse is equal to the sum of the areas of the squares formed on the two legs.

There are many proofs of this important theorem. Figure 5.1 shows a tessellation proof. The plane is tessellated, or tiled,...

9. 6 Elegant Solutions
(pp. 109-130)

What makes a difficult mathematics problem easy? Sometimes there is a sudden flash of understanding. Sometimes past experience points out the right direction to take. This chapter presents problems whose solutions illustrate concepts or techniques that can be appreciated for their power and beauty, and may be useful to you in future problem solving.

Four spheres of radius 1 are contained in a regular tetrahedron in such a way that each is tangent to three faces of the tetrahedron and to the other three spheres. What is the side length of the tetrahedron?

In a regular tetrahedron, letrbe...

10. 7 Creative Problems
(pp. 131-138)

It is easy to formulate new mathematical problems. One only needs an inquiring mind. The problems in this chapter are partially or completely unsolved, so there is much to work on!

Choose a positive integer, say, 20. Now choose a random integer between 1 and 20, say, 9. Subtract: 20 − 9 = 11. Next, choose a random integer between 1 and 11, say, 7. Subtract: 11 − 7 = 4. Choose a random integer between 1 and 4, say, 3. Subtract: 4 − 3 = 1. Now we must choose the integer 1, and we subtract: 1 − 1...

11. A Harmonious Foundations
(pp. 139-150)
12. B Eye-Opening Explorations
(pp. 151-164)
13. Bibliography
(pp. 165-168)
14. Index
(pp. 169-176)