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★ Geometry

Geometry: see Mathematics & Art & Architecture & Sculpture & Drawing & Painting & Arts & Crafts & Music & Science & Fractals & Harmony & Stonehenge

Albrecht Durer - Joshua Reynolds - Plato - Steven Chu - Isaac Newton - Erno Rubik - Pythagoras - Johannes Kepler - Everest Boole - Henri Poincare - David Hilbert - Nicholas Murray Butler - Henry Parker Manning - Eric Temple Bell - Florian Cajori - Bertrand Russell - Benoit Mandelbrot - Michael F Barnsley - Marcus du Sautoy TV - Operation Stonehenge: What Lies Beneath TV -     

 

 

11,159.  Since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art.  (Art & Geometry)  Albrecht Durer, The Art of Measurement 1525

 

 

11,415.  It is the very same taste which relishes a demonstration in geometry, that is pleased with the resemblance of a picture to an original, and touched with the harmony of music.  (Art & Geometry)  Joshua Reynolds, Discourses on Art December 1769

 

 

74,702.  Geometry, rightly treated, is the knowledge of the eternal.  Plato

 

 

74,703.  Geometry was the first exciting course I remember.  Steven Chu

 

 

83,062.  Geometry does not teach us to draw these lines, but requires them to be drawn; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry; then it shows how by these operations problems may be solved.  (Newton & Geometry)  Isaac Newton 

 

 

74,704.  I’ve always been passionate about geometry and the study of three-dimensional forms.  Erno Rubik                  

 

 

74,705.  There is geometry in the humming of the strings, there is music in the spacing of the spheres.  (Geometry & Music)  Pythagoras

 

 

74,706.  Where there is matter, there is geometry.  (Geometry & Matter)  Johannes Kepler

 

 

74,709.  Geometry is one and eternal shining in the mind of God.  That share in it accorded to men is one of the reasons that Man is the image of God.  Johannes Kepler

 

 

74,707.  Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions.  Mary Everest Boole, Lectures on the Logic of Arithmetic 1903 Preface 18

 

 

74,708.  Geometry is not true, it is advantageous.  Henri Poincaré, Science and Hypothesis 1902

 

 

74,710.   Geometry is the most complete science.  David Hilbert

 

 

74,711.  The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method – more daring than anything that the history of philosophy records – of Lobachevsky and Riemann, Gauss and Sylvester.  Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.  Nicholas Murray Butler, presidential address to National Education Association ‘What Knowledge is of Most Worth?’

 

 

74,712.  The greatest advantage to be derived from the study of geometry of more than three dimensions is a real understanding of the great science of geometry.  Our plane and solid geometries are but the beginning of this science.  The four-dimensional geometry is far more extensive than the three-dimensional, and all the higher geometries are more extensive than the lower.  Henry Parker Manning, Geometry of Four Dimensions 1914

 

 

74,713.  The only royal road to elementary geometry is ingenuity.  Eric Temple Bell, The Development of Mathematics

 

 

74,714.  Students in analytical geometry should know something of Descartes, and, after taking up the differential and integral calculus, they should become familiar with the parts that Newton, Leibniz, and Lagrange played in creating that science.  Florian Cajori, A History of Mathematics p4 1893

 

 

74,715.  The influence of geometry upon philosophy and scientific method has been profound.  Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems which are very far from self-evident.  The axioms and theorems are held to be true of actual space, which is something given in experience.  It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction.  This view influenced Plato and Kant, and most of the intermediate philosophers ... The eighteenth century doctrine of natural rights is a search for Euclidean axioms in politics.  The form of Newton's Principia, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source.  Bertrand Russell (1945) A History of Western Philosophy Book 1945

 

 

73,568.   I conceived and developed a new geometry of nature and implemented its use in a number of diverse fields.  It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call fractals.  (Geometry & Fractal)  Benoit Mandelbrot, cited The Fractal Geometry of Nature 1977 introduction

 

74,716.  I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid – a term used in this work to denote all of standard geometry – Nature exhibits not simply a higher degree but an altogether different level of complexity ... The existence of these patterns challenges us to study these forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous.  (Geometry & Fractal)  ibid.  91:9

 

 

74,567.  Fractal geometry will make you see everything differently.  There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides.  Never again will your interpretation of these things be quite the same.  (Geometry & Fractal)  Michael F Barnsley, Fractals Everywhere 2000

 

 

80,677.  This is the Giant’s Causeway at the northern tip of northern Ireland.  And it’s farmed for these strange angular rocks.  There are forty thousand of them ... These stones tell of a hidden geometric force that underpins and pervades all Nature.  And if we can uncover that force it will tell us the shape of everything.  (Maths & Geometry)  Professor Marcus du Sautoy, The Code II: Shapes

 

80,678.  The bees’ honeycomb is a marvel of natural engineering.  The bees have made an identical pattern to the columns on the Giant’s Causeway.  Each cell is exactly like the others ... It’s as if the hexagon is built into the bees’ DNA.  (Maths & Geometry)  ibid.

 

80,679.  It’s the hexagons that use the least amount of wax.  (Maths & Geometry)  ibid.

 

80,680.  This is Nature’s code at work and the bees are in tune with it.  (Maths & Geometry)  ibid.

 

80,681.  The soap bubble reveals something fundamental about Nature – it’s lazy; it tries to find the most efficient shape, the one using the least energy, the least amount of space.  (Maths & Geometry)  ibid.

 

80,682.  Frei Otto started something of a revolution in architecture.  The sweeping curves of the Olympic Stadium are echoed in countless modern structures.  (Maths & Architecture & Geometry)  ibid.

 

80,683.  The mainstay of Greek geometry was the discovery of five perfect shapes now called the Platonic solids ... The tetrahedron with its four faces, the cube with its six faces, the octahedron with its eight faces, the dodecahedron twelve faces, and the most complicated shape of all the icosahedron with its twenty faces.  Today these are more commonly known as dice ... These are the only five shapes like this that can possibly exist; they are the only perfectly symmetrical solids.  (Maths & Geometry)  ibid.

 

80,684.  The world clearly isn’t just built from simple geometric shapes.  (Maths & Geometry)  ibid.   

 

 

89,844.  The monuments’ innate symmetry has revealed that the architects of Stonehenge had a grasp of geometry.  (Stonehenge & Monument & Geometry)  Operation Stonehenge: What Lies Beneath? II