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2015 | Book

Art From Tiling Patterns Read chapter Read first chapter

Treks into Intuitive Geometry

The World of Polygons and Polyhedra

Authors: Jin Akiyama, Kiyoko Matsunaga

Publisher: Springer Japan

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book is written in a style that uncovers the mathematical theories buried in our everyday lives such as examples from patterns that appear in nature, art, and traditional crafts, and in mathematical mechanisms in techniques used by architects. The authors believe that through dialogues between students and mathematicians, readers may discover the processes by which the founders of the theories came to their various conclusions―their trials, errors, tribulations, and triumphs. The goal is for readers to refine their mathematical sense of how to find good questions and how to grapple with these problems. Another aim is to provide enjoyment in the process of applying mathematical rules to beautiful art and design by examples that highlight the wonders and mysteries from our daily lives. To fulfill these aims, this book deals with the latest unique and beautiful results in polygons and polyhedra and the dynamism of geometrical research history that can be found around us. The term "intuitive geometry" was coined by Lászlo Fejes Tóth to refer to the kind of geometry which, in Hilbert's words, can be explained to and appeal to the "man on the street." This book allows people to enjoy intuitive geometry informally and instinctively. It does not require more than a high school level of knowledge but calls for a sense of wonder, intuition, and mathematical maturity.

Table of Contents

Frontmatter
Chapter 1. Art From Tiling Patterns
Abstract
Hi, Kyuta. You look bored.
Jin Akiyama, Kiyoko Matsunaga
Chapter 2. The Tile-Maker Theorem and Its Applications to Art and Designs
Abstract
Hey, Kyuta! Can you search for artworks by Escher on the internet?
Jin Akiyama, Kiyoko Matsunaga
Chapter 3. Patchwork
Abstract
Look, there are a lot of patchwork quilts here (Fig. 3.1.1). Making patchwork quilts is a process of sewing different shapes (and designs) of pieces of cloths together to make one connected cloth, right?
Jin Akiyama, Kiyoko Matsunaga
Chapter 4. Reversible Pairs of Figures
Abstract
I have here two figures: one is a shrimp and the other is a bream (Fig. 4.1.1). In Japan, there is a well-known saying, “throw a shrimp to catch a bream” which has same meaning as “throw a sprat to catch a mackerel” in English.
Jin Akiyama, Kiyoko Matsunaga
Chapter 5. Platonic Solids
Abstract
The ancient Greek philosopher Plato loved five special polyhedral (Fig.5.1.1). These were the solids with equivalent faces composed of congruent regular convex polygons. Today, we know them as Platonic solids; and there are exactly five of them (Fig. 5.1.2).
Jin Akiyama, Kiyoko Matsunaga
Chapter 6. Cross-Sections of Polyhedra
Abstract
Kyuta visits his friend who was in hospital.
When he returns home, Gen was there.
Jin Akiyama, Kiyoko Matsunaga
Chapter 7. Symmetry of Platonic Solids
Abstract
Platonic solids have many beautiful properties, because they are highly symmetric.
Remember the ones we discussed in the previous chapter? Now, I will show you a few more results which relate to the symmetry of Platonic solids.
Let's start with the lengths of diagonals.
Jin Akiyama, Kiyoko Matsunaga
Chapter 8. Double Duty Solids
Abstract
I need a cold drink; would you want one?
Jin Akiyama, Kiyoko Matsunaga
Chapter 9. Nets of Small Solids with Minimum Perimeter Lengths
Abstract
In our city, the waste treatment center from each ward office sends a garbage truck every other day to the houses in the area to collect garbage. On those days, we sort out garbage into recyclable, non-burnable, burnable, etc.; and some of it, like papers and plastic bottles, are recycled. Also, we help the garbage collection ward by minimizing the volume of garbage containers. For example, if we put empty boxes into a garbage bag, we should flatten them to decrease the total volume. So, I now have a common question that we should consider on a daily basis, especially like for garbage. For a given paper polyhedron P, what is the most efficient way to make it flat? That is to say, how can we minimize the total length d(P) (or simply d) of segments along which the surface of P was cut to make a net of P? A net obtained in this manner is called a net with minimum perimeter length (NMPL), or a minimum perimeter net, for short. If we represent the perimeter length of P by ℓ(P), then ℓ(P) = 2d(P) holds.
Jin Akiyama, Kiyoko Matsunaga
Chapter 10. Tessellation Polyhedra
Abstract
Among the infinitely many convex polyhedra that exist, there are some whose faces are all regular polygons. They are called “regular faced polyhedra” (RFP for short). Can you guess how many such polyhedra there are?
Jin Akiyama, Kiyoko Matsunaga
Chapter 11. Universal Measuring Boxes
Abstract
Since the old days of Japan, there were many ingenious convenient tools that applied mathematical ideas such as the right-angled scale (kanejaku), and the measuring box (masu). The kanejaku is a right-angled scale that was used to find the center and diameter of a circle (Fig. 11.1.1 (a)). On the other hand, the masu is a simple wooden box with no markings whose volume is, in this example, 6 dL. Using this box we can measure 1 dL, 2 dL, 3 dL, and so on in units of 1 dL up to 6 dL [1, 4].
Jin Akiyama, Kiyoko Matsunaga
Chapter 12. Wrapping a Box
Abstract
Hi, Kyuta. What are you doing?
Jin Akiyama, Kiyoko Matsunaga
Chapter 13. Bees, Pomegranates and Parallelohedra
Abstract
Let's start our discussion with this box which I intend to fill in with as many cans as possible. Look, it is filled with 40 cans already (Fig. 13.1.1). But I'd still like to add one more can in here [1, 16]. It looks like fully packed already, but can I squeeze in one more can? What do you think?
Jin Akiyama, Kiyoko Matsunaga
Chapter 14. Reversible Polyhedra
Abstract
I have here a pig. Actually, it is a truncated octahedron (Fig. 14.1.1). And I can transform this pig into something else if I turn it inside out. What do you think will I get?
Jin Akiyama, Kiyoko Matsunaga
Chapter 15. Elements of Polygons and Polyhedra
Abstract
In this chapter, I'd like to show you the building blocks for Platonic solids. We can call them ‘elements’, in the same way that in chemistry, materials are made up of, or synthesized from chemical elements. In physics, particle physicists study the elementary particles (physical elements, so to speak), which make up matter.
Jin Akiyama, Kiyoko Matsunaga
Chapter 16. The Pentadron
Abstract
Now let's talk about element sets for the five families of parallelohedra.
Jin Akiyama, Kiyoko Matsunaga
Backmatter
Metadata
Title
Treks into Intuitive Geometry
Authors
Jin Akiyama
Kiyoko Matsunaga
Copyright Year
2015
Publisher
Springer Japan
Electronic ISBN
978-4-431-55843-9
Print ISBN
978-4-431-55841-5
DOI
https://doi.org/10.1007/978-4-431-55843-9

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