Platonic Solids

Platonic Solids

-Describe these objects-

• What are some things that you notice?

• Have you ever seen anything like these? Where?

• What do they remind you of?

• How would you describe these objects?

• How can we describe these using geometric terms?

Hmmmm…. What do you think now?

• http://www.youtube.com/watch?v=gVzu1_12FUc

Table of Contents• Platonic Solids- What are they?• FIVE• Euler’s Formula• History • Traditional Elements• In the Real World• Leonardo DaVinci• Creating & Exploring

The Platonic Solids – What are they?• Platonic solids = Regular polyhedra• Only five polyhedra that are completely regular

convex (all sides are flat) polyhedra • All sides are same length• All angles are the same• All faces are congruent (rotated or flipped but still the

same shape)• Arrangement of faces at each vertex is the same• Only one type of regular polygon is used in each Platonic

solid.

The Five Platonic Solids• Tetrahedron Cube

Octahedron

• Dodecahedron Icosahedron

Euler’s Formula• (# of faces)+ (# of vertices)- (# of edges) = 2• Ex) A cube has 6 faces, 8 vertices and 12 edges

6+8-12=2• You can use this formula to check your work.

• Manipulating the equation:• V=E-F+2 • F= E-V+2• E= F+V-2

• Through extensive proofs, using this formula could prove why only 5 Platonic Solids.

History ~ Platonic Solids• Named after Plato- very impressed • Wrote an entire book on these five solids called

Timaeus• Equated the 5 solids with the 5 traditional

elements: • earth, water, air, fire and universe.

History ~ Platonic Solids- Who’s to credit?• Not known who discovered the Platonic solids but

it is known who the possible contributors were. • Pythagoras and the Pythagoreans with their

discovery?• Only been familiar with the tetrahedron, cube,

and dodecahedron?• Discovery of the octahedron and icosahedron

belong to Theaetetus, a contemporary of Plato?

History ~ Platonic Solids

“It is a fact that approximately 300 examples of the five perfect solids carved from stone have been unearthed in Scotland, whose dates of

creation have been set at more than 1,000 years earlier than Plato- making these solids far from a ‘Platonic discovery’”

• Possibility that Plato learned about these solids from the Pythagoreans who learned about geometry from the Egyptians.

• Should these solids be named after Plato? What’s our point?

• Evidence that earlier cultures discovered before Plato did

Platonic Solids- Traditional Elements• Related to the four traditional elements

-Plato made the following identifications-

Tetrahedron = fire (heat feels sharp and stabbing like tetrahedron)• Octahedron = air (its miniscule components are so bare

that one can barely feel it)• Icosahedron = water (flows out of ones hand when

picked up)• Cube= earth • Dodecahedron = entire universe

Platonic Solids in the Real World

Leonardo DaVinci• Dodecahedron

Let’s Create Some!• Can you create more than five different platonic

solids?• Why or why not?• Turn and talk with a partner

Platonic Solid

Picture Faces

Shape of Faces

Vertices

Edges

Dual

Tetrahedron 4 Triangle 4 6 Tetrahedron

Cube 6 Square 8 12 Octahedron

Octahedron 8 Triangle 6 12 Cube

Dodecahedron

12 Pentagon 20 30 Icosahedron

Icosahedron

20 Triangle 12 30 Dodecahedron

Resources• http://

www.natures-word.com/sacred-geometry/the-platonic-solids/introduction-to-the-platonic-solids

• http://www.enchantedlearning.com/math/geometry/solids/

• http://rchsbowman.wordpress.com/2009/11/18/geometry-notes-history-the-platonic-solids/

• http://www.mathsisfun.com/geometry/platonic-solids-why-five.html

• http://en.wikipedia.org/wiki/Platonic_solid• http://

milan.milanovic.org/math/english/solids/solids.html

• http://www.0disease.com/0platinicsolid_rest.html