Download e-book for iPad: Amazing Math: Introduction to Platonic Solids by Sunil Tanna

Geometry Topology

By Sunil Tanna

This booklet is a advisor to the five Platonic solids (regular tetrahedron, dice, standard octahedron, typical dodecahedron, and typical icosahedron). those solids are very important in arithmetic, in nature, and are the one five convex normal polyhedra that exist.

issues coated contain:

  • What the Platonic solids are
  • The heritage of the invention of Platonic solids
  • The universal positive factors of all Platonic solids
  • The geometrical information of every Platonic reliable
  • Examples of the place each one kind of Platonic reliable happens in nature
  • How we all know there are just 5 different types of Platonic sturdy (geometric facts)
  • A topological facts that there are just 5 varieties of Platonic strong
  • What are twin polyhedrons
  • What is the twin polyhedron for every of the Platonic solids
  • The relationships among each one Platonic reliable and its twin polyhedron
  • How to calculate angles in Platonic solids utilizing trigonometric formulae
  • The courting among spheres and Platonic solids
  • How to calculate the skin quarter of a Platonic strong
  • How to calculate the amount of a Platonic good

additionally incorporated is a quick creation to a couple different fascinating sorts of polyhedra – prisms, antiprisms, Kepler-Poinsot polyhedra, Archimedean solids, Catalan solids, Johnson solids, and deltahedra.

a few familiarity with easy trigonometry and intensely simple algebra (high institution point) will let you get the main out of this ebook - yet for you to make this publication available to as many folks as attainable, it does comprise a short recap on a few worthy simple suggestions from trigonometry.

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Additional resources for Amazing Math: Introduction to Platonic Solids

Sample text

The Catalan solids are named after the Belgian mathematician Eugène Catalan (May 30th, 1814 to February 14th, 1894) who first described them in 1865. Here are the Catalan solids: Here are the names of the Catalan solids (in the same order as the illustration above, starting from the top-left, and then horizontally across each row, then vertically): Triakis hexahedron Triakis octahedron Rhombic dodecahedron – Note: you may recall from earlier in this book, that when people say that a diamond or garnet exhibits "dodecahedral habit", they are referring to this shape.

However, each edge actually has two ends, and thus contributes to two vertices, so you can double the number giving ( 2E ÷ q ). We can now substitute the formulas for F and V into the earlier equation for χ, giving: Or just: Rearranging this equation leads us to: Now, since we know that E is always positive, this means that is also always positive. We can therefore reach this inequality: Additionally, we know that p and q must both be positive integers (whole numbers) greater than or equal to 3.

These relationships between each polyhedron and its respective dual are shown in the following tables: Calculating Angles in Platonic Solids In this chapter I will talk about some of the angular properties of Platonic solids (I have previously simply given figures for some of the angles without explanation). The mathematics involved is slightly more complex that in most other parts of this book, so before doing so I will briefly explain some necessary mathematical terminology and concepts – however should you wish to study these concepts in more depth, I would suggest that you do some further reading on trigonometry.

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