Amazing Math: Introduction to Platonic Solids by Sunil Tanna

By Sunil Tanna

This booklet is a consultant to the five Platonic solids (regular tetrahedron, dice, ordinary octahedron, commonplace dodecahedron, and average icosahedron). those solids are very important in arithmetic, in nature, and are the single five convex standard polyhedra that exist.

subject matters coated comprise:

  • What the Platonic solids are
  • The historical past of the invention of Platonic solids
  • The universal positive aspects of all Platonic solids
  • The geometrical information of every Platonic stable
  • Examples of the place each one form of Platonic strong happens in nature
  • How we all know there are just 5 forms of Platonic sturdy (geometric evidence)
  • A topological evidence that there are just 5 varieties of Platonic sturdy
  • What are twin polyhedrons
  • What is the twin polyhedron for every of the Platonic solids
  • The relationships among each one Platonic stable 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 zone of a Platonic stable
  • How to calculate the quantity of a Platonic stable

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

a few familiarity with easy trigonometry and intensely easy algebra (high institution point) will let you get the main out of this booklet - yet as a way to make this e-book obtainable to as many folks as attainable, it does contain a quick recap on a few precious easy options from trigonometry.

Show description

Read or Download Amazing Math: Introduction to Platonic Solids PDF

Best geometry & topology books

Local and Analytic Cyclic Homology (EMS Tracts in Mathematics)

Periodic cyclic homology is a homology concept for non-commutative algebras that performs an analogous position in non-commutative geometry as de Rham cohomology for tender manifolds. whereas it produces reliable effects for algebras of delicate or polynomial capabilities, it fails for greater algebras corresponding to such a lot Banach algebras or C*-algebras.

Geometry. A comprehensive course

"A lucid and masterly survey. " — arithmetic GazetteProfessor Pedoe is well known as a superb instructor and a great geometer. His skills in either parts are sincerely obtrusive during this self-contained, well-written, and lucid creation to the scope and strategies of uncomplicated geometry. It covers the geometry often incorporated in undergraduate classes in arithmetic, apart from the idea of convex units.

Foundations of Geometry

The fabric inside the following translation used to be given in substance through Professor Hilbert as a process lectures on euclidean geometry on the college of Göttingen in the course of the iciness semester of 1898–1899. the result of his research have been re-arranged and positioned into the shape during which they seem the following as a memorial tackle released in reference to the party on the unveiling of the Gauss-Weber monument at Göttingen, in June, 1899.

Calculus Revisited

During this ebook the main points of many calculations are supplied for entry to paintings in quantum teams, algebraic differential calculus, noncommutative geometry, fuzzy physics, discrete geometry, gauge idea, quantum integrable platforms, braiding, finite topological areas, a few points of geometry and quantum mechanics and gravity.

Additional resources for Amazing Math: Introduction to Platonic Solids

Sample text

3) Each polygonal face must contribute an equal number of degrees to the junction at a vertex of a polyhedron. The number of degrees that each face contributes to the junction is given by the angle of the vertices within the polygon. (4) Because we want a convex shape, the total number of degrees in the vertices of the polygons meeting at a junction must be less than 360°. Note: It can not be exactly 360° since this would result in a planar surface (flat surface). (5) An equilateral triangle (3-sided regular polygon) has a 60° angles at its vertices, and so the possible combinations around a polyhedron's vertex are 3 equilateral triangles (resulting in a tetrahedron), 4 equilateral triangles (resulting in an octahedron), or 5 equilateral triangles (resulting in an icosahedron).

Here is a net (unfolded version) of an octahedron: Regular Octahedra in Nature Like tetrahedral structures, octahedral structures also occur in chemistry. Just as a central atom surrounded by four atoms/groups can form a tetrahedral structure (albeit sometimes distorted if the atoms/groups are not all the same), a central atom surrounded by six atoms/groups will form an octahedral structure. As with the tetrahedral structure, this happens because the surrounding atoms/groups mutually repel, and hence are evenly spaced as far apart as possible, at the octahedron's vertices.

There are 30 vertices in a regular dodecahedron, each vertex being formed where 3 faces meet. There are 20 edges (formed whenever only 2 faces meet) in a regular dodecahedron. The face angle (the angle at each vertex on each polygonal face) is 108°. 57° (approximately). The vertex angle (the angle between edges at a vertex) is 108°. Here is a net (unfolded version) of a dodecahedron: Dodecahedra in Nature Compared to the other Platonic solids, the dodecahedral shape occurs relatively infrequently in nature.

Download PDF sample

Rated 4.63 of 5 – based on 40 votes