Summary Õ PDF, DOC, TXT, eBook or Kindle ePUB free ¹ Ian Stewart
From the shapes of clouds to dewdrops on a spider's web this accessible book employs the mathematical co There s a fairly high probability that anyone who s interested in mathematics either professionally or simply out of curiosity has read one or books by Ian Stewart He s an accomplished professional mathematician with interests in many branches of the subject But he s also a very good writer and expositor of diverse mathematical topics Most of his books have been intended for general readers with an interest in mathematics These include popular topics like chaos theory the mathematics of biology mathematical recreations and mathematical curiosities His Wikipedia page lists about 36 books of this type But he s also written well about advanced topics such as Galois theory and algebraic number theoryOne of Stewart s particular interests is symmetry which makes a lot of sense because symmetry is a pervasive theme in mathematics and has been for some time Mathematically symmetry is usually studied in terms of transformations that can be performed on an object and which leave the object essentially the same in some sense Plane geometry or geometry in any number of dimensions for that matter provides some of the most obvious examples Consider an euilateral triangle all of whose sides are the same length Such a triangle can be rotated about its center in either 120 or 240 degrees and will look identical to how it did before rotation Mathematically a non rotation i e by 0 degrees is also considered and trivially doesn t change a thing The triangle can also be reflected across a line from any vertex to the middle of the opposite side This is known naturally as reflection symmetry Clearly too it makes no change at all to the triangle s appearanceHowever symmetry can also be considered in a general sense in which a transformation does make some change to the appearance of an euilateral triangle yet still leave it the same in a relaxed sense A transformation that only moves the triangle from one place to another changes nothing but the position yet is still considered a transformation one which could be applied to almost anything not just an euilateral triangle For example a wallpaper design A rotation about the center by any number of degrees can be considered a symmetry of the triangle if you don t care about the direction the vertices point with respect to anything else in the plane Expansion or contraction of the triangle changes only its size and that doesn t affect any of its abstract geometric properties such as the fact that the sum of the interior angles of the vertices is always 180 degrees That s true for any type of triangle with straight sides not necessarily an euilateral oneIt s possible to make even drastic transformations to a triangle which can also be considered symmetries in a looser sense For instance the lengths of two sides could be doubled or multiplied by any positive number but the object would still be a triangle with many properties unchanged such as the sum its interior angles or simply continuing to be a 3 sided polygon Even interestingly transformations can be combined with each other and still yield a symmetry in the abstract sense A certain limited number of combined transformations may still leave an euilateral triangle completely unchanged reflections and rotations by multiples of 120 degrees for example Sets of transformations that can be combined with each other are very important mathematical objects themselves known as groups Stewart devotes the second chapter to a brief discussion of group theory but doesn t say much about abstract groups in the remainder of the book Groups are very interesting but their theory can uickly become uite complicated and deep Group theory is a pervasive topic in modern mathematics and has been for almost 200 years Stewart doesn t go further with the subject or its history in the present book but does in a later book Why Beauty is TruthInstead most of the book presently under discussion is concerned with cases where symmetry is broken that is when the appearance or some other aspect of a real world object becomes different in some way The object no longer has a perfect symmetry of some sort but only one that is merely approximate generally for understandable reasons Consider an automobile practically any kind Almost all cars have a left right mirror symmetry which is a reflection in a vertical plane that runs through the center of the car from the front to the back But it s not perfect because the steering wheel and various switches and gauges on the dashboard are only on one side or the other That s because hardly any cars now provide for anything except one single driver In addition the layout of things under the hood is usually not symmetric either since most cars have only one battery alternator etc Humans and most other higher animals don t have perfect left right symmetry either for similar reasons They typically have only one heart liver stomach etc not in the exact center of the body even though a few organs like kidneys and lungs are duplicated symmetrically and the redundancy is useful In principle a car could be built that had nearly perfect bilateral symmetry The seats for driver and passengers could be located in a straight line from front to back as with fighter planes with than one occupant But the vehicle would then probably be much narrower to avoid wasted space and conseuently likely to tip over when making sharp turns So the upshot is that nature often finds it convenient to break perfect symmetries even if only slightlyMost of the rest of Stewart s book goes into many examples of this symmetry breaking For instance although most land animals with legs have externally at least bilateral symmetry they are mostly unable to move at all unless the legs don t all move the same way at the same time Hopping animals like kangaroos are an exception of course There s a whole chapter in the book about the different ways that animals move their legs in order to walk or run and the pattern may change depending on how fast the animal needs or wants to goThere s a whole chapter on the symmetry of crystals A fascination with crystals whether or not taken to unreasonable extremes has been common in humans from prehistoric times even before homo sapiens as Stewart points out All crystals have some sort of symmetry at least in principle though often broken because of how their constituent atoms arrange themselves The study of crystal symmetries was historically almost the earliest example of the use of group theory in science Different minerals are in fact characterized by the types of symmetries their crystals may exhibit The symmetries are usually slightly broken due to asymmetries of the chemical environment in which the crystal grew Randomness is pervasive in nature but often leads to its own symmetries such as the way that smoke entering one side of a room will uickly become distributed fairly evenly within the roomNatural laws such a gravity have symmetries too A rock tossed upwards at an angle will follow a symmetric parabola A deep theorem due to the mathematician Emmy Noether explains how conservation laws such conservation of energy and momentum which govern the motion of tossed rocks are a result of symmetries in the euations of physicsCrystals aren t alive but animals are and they have symmetries too even though usually only approximate as noted above Consider the stripes of a tiger They are arranged albeit rather irregularly in a periodic seuence along the length of the animal s body literally from head to tail It turns out that this symmetry is a lot like the symmetry of ocean waves approaching a beach Periodic phenomena have deep mathematical generalizations harmonic analysis in which group theory is of central importance Consider how combinations of different freuencies of sound waves are what constitute music Animal patterns such as tiger stripes result from waves of chemical densities in the liuid medium inside very young embryos because they affect gene expression Alan Turing the early computer scientist who conceived Turing machines was apparently the first to suggest this even before genes were actually understood This story is told in another chapter of Stewart s book
Review Fearful Symmetry Is God a Geometer?
Ncepts of symmetry to portray fascinating facets of the physical and biological world More than 120 figu Nice book easily accessible even to people with no mathematical background and with very interesting insights and very good explanations and examples of what is symmetry and symmetry breaking I thoroughly enjoyed reading this book The only small defect is that the influence of symmetry and symmetry breaking in particle physics and in uantum mechanics where these concepts play a huge role was addressed only uite superficially
Summary Õ PDF, DOC, TXT, eBook or Kindle ePUB free ¹ Ian Stewart
Res illustrate the interaction of symmetry with dynamics and the mathematical unity of nature's patterns Part of my job includes teaching uantum mechanics to other people so when I find a math book that not only offers up insightful takes on mathematical concepts I m not already nauseatingly familiar with but does so in humorous and accessible prose I get truly excited Not only have I enhanced my own learning I ve found a teaching example to look up to That said not even Ian Stewart could make me care about the biology sections
- Fearful Symmetry Is God a Geometer?
- Ian Stewart
- 23 February 2019 Ian Stewart