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