© Copyright Lorena Loo
Text and images (unless otherwise indicated in the credits) are copyrighted
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
Flatland to Sphereland & Beyond
Part III Continued
©Lorena Loo
Part III Continued
©Lorena Loo
Dodeca the Universe (Continued)
From our outer and middle 3 concentric phi cubes, we formed the dodecahedron and the middle cube is exactly the cube nested inside the dodeca. If we now form an icosahedron from the inner of the 3 concentric phi cubes, we will get an icosahedron nested inside the octahedron nested inside the tetrahedron nested inside the cube nested inside the dodecahedron. There is just one caveat when we form this icosa to ensure proper nesting inside the octahedron. The 3 pairs of parallel lines formed on the opposite faces of the cube to form the icosa must be perpendicular to those on the corresponding faces of the outer cube used to form the dodeca. When that happens, there is magic. The icosa not only perfectly nests within the octa, each of its 12 vertices phi cuts an edge of the octahedron (which has 12 edges).
From our outer and middle 3 concentric phi cubes, we formed the dodecahedron and the middle cube is exactly the cube nested inside the dodeca. If we now form an icosahedron from the inner of the 3 concentric phi cubes, we will get an icosahedron nested inside the octahedron nested inside the tetrahedron nested inside the cube nested inside the dodecahedron. There is just one caveat when we form this icosa to ensure proper nesting inside the octahedron. The 3 pairs of parallel lines formed on the opposite faces of the cube to form the icosa must be perpendicular to those on the corresponding faces of the outer cube used to form the dodeca. When that happens, there is magic. The icosa not only perfectly nests within the octa, each of its 12 vertices phi cuts an edge of the octahedron (which has 12 edges).
Above left: Unique inward nesting sequence of the Platonic solids. Dodeca (gold), cube (blue), tetra (pink), octa (orange), icosahedron (green).
Above right: 2D net for tetra (pink) nesting octa (orange) where the colored dots represent the vertices of the icosa nested inside the octa. Each of those dots precisely phi cuts an edge of the octa.
Above right: 2D net for tetra (pink) nesting octa (orange) where the colored dots represent the vertices of the icosa nested inside the octa. Each of those dots precisely phi cuts an edge of the octa.
If we take the cube to have unit edge lengths, then the lengths of the edges of the Platonic solids in this special 5 unit nested set are:
Dodecahedron........ Octahedron.............
Cube..................... 1
Tetrahedron............ Icosahedron............
The dodeca's edge lengths then are phi times that of the icosa's edge lengths. In the next section, we will see there is even more to this golden relationship between this dodeca and icosa.
Now imagine the sphere which just encloses this entire nested set of Platonic solids. The vertices of the dodecahedron will lie on the surface of this sphere which is called the circumsphere of the dodeca. All the vertices of the cube and the tetra will also lie on this sphere so the circumsphere of the dodeca is also that of its nested cube and tetrahedron. This is the sphere which Philolaus referred as containing within it the 5 elements.
The dodeca, the fifth element and the hull of the sphere, is the most "spherical" of the 5 elements or Platonic solids. In the Timaeus, Plato said God used the fifth for the whole, placing a pattern of animal figures on it. Plato meant the signs of the zodiac with one sign per pentagonal face. If you have 12 regular pentagons which are rigid, you can construct a dodecahedron. But if they are flexible (like pieces of leather), you can expand the dodeca into a spherical shape to get a spherical dodecahedron like the stone spherical dodecahedrons created by Neolithic people.
Incidentally, you can find the square roots of 2 and 3 easily within the dodeca because of its nested cube. Recall the square root of 2 is the diagonal of the face of a cube and the square root of 3 the diagonal of the cube.
Golden Secret of the Dodeca
Before we proceed any further-and we are going to cover a lot in this section-this is a good place to explain what a Platonic solid is.
A 3-dimensional object (solid) whose faces are all the same regular polygon and which has the same number of faces meeting at each vertex is called a Platonic soild. For a Platonic solid, all its edges will be of the same length and all its angles are equal. (For those unfamiliar with the term regular polygon, I've written an easy to understand definition of it in What Is a Pyramid.) The tetrahedron, octahedron and icosahedron all have equilateral (all sides of equal length) triangular faces with 3, 4, and 5 faces meeting at any vertex respectively. The cubes has square faces with 3 faces meeting at any vertex and the dodeca has regular pentagonal faces with 5 faces meeting at any vertex.
There are only 5 solids which meet the criteria of being a Platonic solid and 5 or fiveness is something which we will come across time and again in this section.
Dodecahedron........ Octahedron.............
Cube..................... 1
Tetrahedron............ Icosahedron............
The dodeca's edge lengths then are phi times that of the icosa's edge lengths. In the next section, we will see there is even more to this golden relationship between this dodeca and icosa.
Now imagine the sphere which just encloses this entire nested set of Platonic solids. The vertices of the dodecahedron will lie on the surface of this sphere which is called the circumsphere of the dodeca. All the vertices of the cube and the tetra will also lie on this sphere so the circumsphere of the dodeca is also that of its nested cube and tetrahedron. This is the sphere which Philolaus referred as containing within it the 5 elements.
The dodeca, the fifth element and the hull of the sphere, is the most "spherical" of the 5 elements or Platonic solids. In the Timaeus, Plato said God used the fifth for the whole, placing a pattern of animal figures on it. Plato meant the signs of the zodiac with one sign per pentagonal face. If you have 12 regular pentagons which are rigid, you can construct a dodecahedron. But if they are flexible (like pieces of leather), you can expand the dodeca into a spherical shape to get a spherical dodecahedron like the stone spherical dodecahedrons created by Neolithic people.
Incidentally, you can find the square roots of 2 and 3 easily within the dodeca because of its nested cube. Recall the square root of 2 is the diagonal of the face of a cube and the square root of 3 the diagonal of the cube.
Golden Secret of the Dodeca
Before we proceed any further-and we are going to cover a lot in this section-this is a good place to explain what a Platonic solid is.
A 3-dimensional object (solid) whose faces are all the same regular polygon and which has the same number of faces meeting at each vertex is called a Platonic soild. For a Platonic solid, all its edges will be of the same length and all its angles are equal. (For those unfamiliar with the term regular polygon, I've written an easy to understand definition of it in What Is a Pyramid.) The tetrahedron, octahedron and icosahedron all have equilateral (all sides of equal length) triangular faces with 3, 4, and 5 faces meeting at any vertex respectively. The cubes has square faces with 3 faces meeting at any vertex and the dodeca has regular pentagonal faces with 5 faces meeting at any vertex.
There are only 5 solids which meet the criteria of being a Platonic solid and 5 or fiveness is something which we will come across time and again in this section.
Animation showing alignment of parallel lines drawn on opposite faces of the outer and inner phi cubes to form the dodeca and icosa respectively.
Now we can return to the unique nesting unit of the 5 Platonic solids. If we shrink the entire unit of 5 by a factor of 3, the dodecahedron of the shrunken nested unit will nest perfectly within the icosa of the original unit of 5. That shrunken dodeca will also be the dual of the original icosa. If we expand the entire unit of 5 Platonic solids by a factor of 3, the dodeca of the original unit will nest inside the icosa of the expanded unit of 5 and will be its dual.
Below is another image and view of this 5 unit Platonic nesting created using vZome. Zome has an actual kit called Kepler's Obsession of this specific nesting set. As I am still waiting to receive my parts and kit from Zome, Scott Vorthmann's wonderful virtual Zome program enabled me to create the model below. Thanks, Scott. And thanks, Paul Hildebrandt for creating Zome.
Below is another image and view of this 5 unit Platonic nesting created using vZome. Zome has an actual kit called Kepler's Obsession of this specific nesting set. As I am still waiting to receive my parts and kit from Zome, Scott Vorthmann's wonderful virtual Zome program enabled me to create the model below. Thanks, Scott. And thanks, Paul Hildebrandt for creating Zome.
The 5 Platonic solids and their duals depicted as wireframe models
This unique nesting sequence of the Platonic solids gives rise to something special when we shrink or grow the entire unit by a factor of 3. Before we go further with that, let's briefly touch upon the concept of the dual of a Platonic solid.
Each Platonic soild has a dual which is formed by joining the centers of the faces of a given Platonic soild. Those centers happen to form the vertices of the solid's dual and its dual is also a Platonic solid. So as an icosahedron has 12 faces, its dual has 12 vertices which is the dodeca. A cube has 6 faces so its dual has 6 vertices which is the octa. A tetrahedron has 4 faces, its dual has 4 vertices which means the tetra is its own dual.
Each Platonic soild has a dual which is formed by joining the centers of the faces of a given Platonic soild. Those centers happen to form the vertices of the solid's dual and its dual is also a Platonic solid. So as an icosahedron has 12 faces, its dual has 12 vertices which is the dodeca. A cube has 6 faces so its dual has 6 vertices which is the octa. A tetrahedron has 4 faces, its dual has 4 vertices which means the tetra is its own dual.
The astronomer Johannes Kepler (1571-1630), who devised the three laws of planetary motion, believed divine geometry was the basis of the structure of the cosmos. Above are the 5 Platonic solids as he depicted them in his first book, Mysterium Cosmographicum (Cosmographic Mystery). The faces of each solid are decorated with images pertinent to the "element" associated with it.
Two dimensionally, the regular polygon associated with fiveness is the pentagon. The pentagon naturally gives rise to the 5-pointed star figure called the pentagram which is all about fiveness and phi-ness. The lines forming the pentagram intersect in such a way that the ratio of a given line segment to the next successively shorter line segment (to it) is phi or the golden ratio.
If you read geometry books by mathematicians, you will discover they invariably use the the Greek letter tau ( ) to designate the golden section. This can often be confusing because they use the value of 0.618033988 for tau as opposed to 1.618033988 which is the value of phi. Their value of tau is
actually .
If you read geometry books by mathematicians, you will discover they invariably use the the Greek letter tau ( ) to designate the golden section. This can often be confusing because they use the value of 0.618033988 for tau as opposed to 1.618033988 which is the value of phi. Their value of tau is
actually .
We can also form a pentagram by extending the sides of a pentagon outwards until they intersect. Instead of being contained within the pentagon, this pentagram contains the original pentagon. By joining the vertices of the pentagram, we can form a larger pentagon whose sides are phi times that of the original pentagon.
The Pythagoreans knew of the relationship between the pentagon and the pentagram. They employed the pentagram as the symbol of their society, regarding it as a symbol of health or
The Pythagoreans knew of the relationship between the pentagon and the pentagram. They employed the pentagram as the symbol of their society, regarding it as a symbol of health or
The center of a pentagon also happens to the center of the circumcircle of the pentagon. By forming a diagonal of the pentagon and rotating or ratcheting it through successive 72º about the center, you generate the other diagonals of the pentagon and hence a pentagram.
The animation to the left demonstrates this using Agrippa's pentagram man from The Circle and the Pentagon. The red dot is the center of the pentagon and also its circumcircle.
This 2-D concept will translate into 3-D
The animation to the left demonstrates this using Agrippa's pentagram man from The Circle and the Pentagon. The red dot is the center of the pentagon and also its circumcircle.
This 2-D concept will translate into 3-D
with a very interesting result.
The above animation illustrates how a pentagram can be formed from the diagonals of a regular pentagon. Contained within the pentagram is another regular pentagon whose sides are phi times smaller than the original pentagon.
The above animation illustrates how a pentagram can be formed from the diagonals of a regular pentagon. Contained within the pentagram is another regular pentagon whose sides are phi times smaller than the original pentagon.
perhaps whole as etymologically health means the state of being whole.
The Pythagoreans held a special reverence for the dodecahedron, the only Platonic solid comprised of regular pentagonal faces. We can only speculate as to what they knew and understood about the dodeca that they regarded it so highly.
We already know we can nest a cube inside the dodeca as illustrated in the photo to the right. The view is looking right down onto a face of the dodeca. An edge of the cube forms a diagonal of
The Pythagoreans held a special reverence for the dodecahedron, the only Platonic solid comprised of regular pentagonal faces. We can only speculate as to what they knew and understood about the dodeca that they regarded it so highly.
We already know we can nest a cube inside the dodeca as illustrated in the photo to the right. The view is looking right down onto a face of the dodeca. An edge of the cube forms a diagonal of
that face. In fact, each edge of the cube is a diagonal of one of the dodeca's pentagonal faces. Take any edge of the cube and ratchet it through successive 72º rotations about the center of the pentagonal face of which it is a diagonal. That racheting will rotate the entire cube as well. Five successive ratchets produces the result seen in the photo below.
This ratcheting of the cube's edge about the center of the face of the dodeca of which it is a diagonal is the same as ratcheting the entire cube about an axis that passes through the centers of opposite faces of the dodeca. This produces 5 unique cubes, each nested within the same dodeca. Eight of the dodeca's 12 vertices also form the vertices of these cubes. Each edge of these cubes forms a diagonal of a pentagonal face of the dodeca so that a pentagram is formed on each face of the dodeca. Now we see fiveness 3 dimensionally and we will also see this
fiveness is phi-ness. It is very probable the Pythagoreans knew how the 5 cubes nested inside the dodeca to produce pentagrams on the faces of the dodeca. But maybe they were not the only ones.