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## Über dieses Buch

Based on a series of lectures for adult students, this lively and entertaining book proves that, far from being a dusty, dull subject, geometry is in fact full of beauty and fascination. The author's infectious enthusiasm is put to use in explaining many of the key concepts in the field, starting with the Golden Number and taking the reader on a geometrical journey via Shapes and Solids, through the Fourth Dimension, finishing up with Einstein's Theories of Relativity.

Equally suitable as a gift for a youngster or as a nostalgic journey back into the world of mathematics for older readers, John Barnes' book is the perfect antidote for anyone whose maths lessons at school are a source of painful memories. Where once geometry was a source of confusion and frustration, Barnes brings enlightenment and entertainment.

In this second edition, stimulated by recent lectures at Oxford, further material and extra illustrations have been added on many topics including Coloured Cubes, Chaos and Crystals.

## Inhaltsverzeichnis

### 1. The Golden Number

THIS LECTURE is about the so-called Golden Number. This number defines a ratio which turns up in various guises in an amazing number of contexts. These include the ideal shapes of architectural objects such as temples, the arrangement of many botanical systems, and purely geometrical objects such as the pentagon. We shall start, however, by considering the sizes of pieces of paper.
John Barnes

### 2. Shapes and Solids

THIS LECTURE is about the variety of regular shapes in two and three dimensions. We start by considering the regular plane figures such as the triangle, square, and pentagon and how they can be used to form various regular patterns of tiles. We then move into the third dimension and consider the simple solid figures, such as the tetrahedron and cube which were known in classical times. We conclude by looking at some of the many more elaborate and beautiful figures discovered more recently.
John Barnes

### 3. The Fourth Dimension

THE PREVIOUS LECTURE described the regular figures in two and three dimensions. We now consider the regular figures in four dimensions and how we can get a glimpse of them by analogy with how a person in Flatland might get some appreciation of figures in three dimensions.
John Barnes

### 4. Projective Geometry

THE GEOMETRY we did at school (or maybe didn’t) was mostly very dull. It was all about lengths and angles; proofs were often about showing that certain angles or lengths were equal. However, there is much geometry in which the lengths of lines and sizes of angles are not considered at all. This so-called projective geometry was heavily studied in the nineteenth century but became unfashionable. This was perhaps because it seemed to have no practical value and did not provide a foundation for other things. Nevertheless, it has a certain elegance and beauty.
John Barnes

### 5. Topology

TOPOLOGY is one of those branches of mathematics where some of the simple results are very easy to state and appreciate. However, the underlying mathematics is rather hard and it is not easy to prove what seems quite obvious. Topology used to be considered a branch of geometry although now it is seen more as a branch of abstract algebra. In this lecture we will just look at some of the entertaining facts and not generally bother with how they might be proved.
John Barnes

### 6. Bubbles

TOAP BUBBLES may seem an odd topic for a lecture. One might think that there was little to say since they are patently spherical and surely that is that. However, when two or more bubbles are joined together some intriguing features are revealed. This lecture is partly inspired by Boys’ famous book entitled Soap Bubbles whose second edition was published in 1911.
John Barnes

### 7. Harmony of the Spheres

IN THE LECTURE on Bubbles we saw how the centres of bubbles, the centres of curvature of dividing films, and points on tangents formed various harmonic ranges. In this lecture we will start by looking at some other pretty properties of circles and spheres. But first a brief explanation of the title of this lecture.
John Barnes

### 8. Chaos and Fractals

THE TWENTIETH CENTURY saw an upheaval in our understanding of the mechanics of the universe and the foundations of mathematics. This lecture looks at two aspects of strange modern mathematics with a clear geometrical interpretation and the subsequent lecture looks at aspects of our understanding of the physical universe as modified by Einstein’s theories of relativity.
John Barnes

### 9. Relativity

EINSTEIN PROPOSED two theories of relativity at the start of the twentieth century which painted a picture of the world quite at odds with our intuition regarding the nature of space and time. The two theories are known as the Special Theory and the General Theory. The Special Theory essentially concerns the velocity of light and the mathematics is fairly easy to understand even though the outcomes are quite startling. The General Theory concerns gravitation and the mathematics behind it is considered rather difficult (postgraduate in these dumbed-down days I am sure). The effects of both theories differ little from the traditional predictions of Newton except in extreme circumstances. Nevertheless, in those extreme circumstances all experiments have indicated that they are correct. This lecture gives a reasonable presentation of the Special Theory and explains phenomenon such as time and space contraction. It also gives just a glimpse of the General Theory which lies behind such things as Black Holes.
John Barnes

### 10. Finale

THESE LECTURES have had a number of goals. One was simply to present some pretty or surprising configurations. Another was to reveal that despite the fact that we live in a three-dimensional world, nevertheless our understanding of three dimensions is fairly poor. Thus few people know that if you cut through a cube in a certain way, then the cross-section is a hexagon. Another goal, and perhaps the most important in a philosophical sense, was that solving and understanding a problem depends very much upon getting the right point of view. The lecture on inversion and its use to prove Steiner’s porism and explain Soddy’s hexlet is perhaps the most intriguing example of getting the right point of view that we have encountered. Another example is that of the train marking the track in special relativity where it is important to analyse the situation from the point of view of the correct observer. In this final lecture we will look at some more examples where getting the right point of view is so important. These are about certain curious properties of triangles, squares and other rectilinear figures and how the use of the Argand plane can provide very simple explanations in many cases.
John Barnes

### Backmatter

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