Platonic solids, Geodesic Dome

Platonic solids

Geodesic Dome

INDEX

General information

Leonardo and Platonic solids

History

Tetrahedron

Cube

Octahedron

Dodecahedron

Icosahedron

Origami

Geodesic Dome

General information

A Platonic solid is a polyhedron where every face is a regular polygon with the same number of edges, and where the same number of faces meet at every vertex. Platonic Solids are the most regular polyhedra: all faces are the same regular polygon, and they look the same at every vertex. There are only five different Platonic solids: the Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron.

History

The original discovery of the platonic solids is unknown. The five regular polyhedra all appear in nature whether in crystals or in living beings. Around 360 B.C. the regular polyhedra are discussed in the dialogues of Plato, their namesake. Because of Plato's systematic development of a theory of the universe based on the five regular polyhedra, they became known as the Platonic solids.

History

Plato discovered that there are only five solids with these properties. He believed that they correspond to the four ancient Elements, as well as the Universe. Fire is associated with the tetrahedron, earth with the cube, air with the octahedron, and water with the icosahedron. Lastly, the dodecahedron is considered to represent the constellations that make up the universe.

History

Similar to Plato, the German astronomer Johannes Kepler also searched for connections between the regular polyhedra and the natural world. He proposed his model for our solar system. At the time they knew of five planets other than earth. Kepler predicted connections between these five planets and the five Platonic solids.

Kepler’s model of the solar system using the Platonic solids

LEONARDO and Platonic solids

Many artists, such as Leonardo da Vinci, as well as and mathematician and Leonardo's friend, Luca Pacoli, used the Platonic Solids in their works as they were interested in the math behind them and the symmetry shown in the solids. Leonardo drew the illustrations for Luca Pacioli's book De Divina Proportione (The Divine Proportion).

Tetrahedron

The Tetrahedron has four triangular faces and is the smallest Platonic solid. It has 7 axes of symmetry. Plato believed it represents the ancient element Fire. Many molecules have their atoms arranged as a Tetrahedron.

hexahedron (Cube)

The Cube has six quadratic faces and 13 axes of symmetry. Plato believed it represents the ancient element Earth. It is often used for dice. Since it is a “regular” solid, every side has the same probability of landing face up.

Octahedron

The Octahedron has eight triangular faces. It is the “dual” solid of the cube. Plato believed it represents the ancient element Air. Many natural crystals are based on an octahedral lattice – including diamond, alum or fluorite.

Dodecahedron

The Dodecahedron has 12 pentagonal faces and 31 axes of symmetry. Plato believed that the entire Universe has the shape of a Dodecahedron.

Icosahedron

The Icosahedron is the largest Platonic solid and has 20 triangular faces. It is the dual of the Dodecahedron. Plato believed that it represents the element Water. In fact, many viruses, such as herpes, have icosahedral shells.

Examples of origami platonic solids

Geodesic dome

Have you ever seen a geodesic dome? Geodesic domes are sphere like structures made up of interconnected triangles. A famous geodesic dome is Walt Disney World’s Spaceship Earth at Epcot, but geodesic domes are also commonly found as climbing domes at playgrounds.

HISTORY

Facts

1. A Geodesic Dome is a curved, three-dimensional structure formed through a network of triangles. The more complex this network of triangles becomes, the closer it begins to approximate the geometry of a true sphere, or any fraction of one. The word Geodesic comes from the Greek root geodaisia, meaning “division of the Earth.

2. The more triangles that are used in a dome, the rounder it becomes. The frequency of a dome indiciates this relationship such that the higher the number, the rounder the surface.

3. Because hot air rises, warm air inside of a dome can create a rising effect similar to that of a hot-air balloon. This phenomenon can actually lift the dome enough to noticeably change the weight of the entire structure. Larger domes that enclose more hot air experience a stronger lifting force.

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