© 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.
Part III
Dodeca the Universe
A dodecahedron has 12 regular (all sides of equal length) pentagonal faces with 20 vertices and 30 edges. Below left is a Zome model of a dodecahedron.
A dodecahedron has 12 regular (all sides of equal length) pentagonal faces with 20 vertices and 30 edges. Below left is a Zome model of a dodecahedron.
The edge lengths of the icosahedron nested within a cube of edge lengths 1 is
.
.
Now we know how a cube nests inside the dodecahedron and we already know how a tetrahedron nests inside a cube. In fact, the tetrahedron which nests inside a cube will also nest inside a dodecahedron. Remember the dodeca "shares" 8 of its 20 vertices with that of the cube it nests. Since 4 of the cube's 8 vertices are the vertices of the tetrahedron it nests, then 4 of the dodeca's vertices are vertices of the tetrahedron. We also know how to form the octahedron which nests inside the tetrahedron (octa also nests inside cube nesting the tetra). There is only one Platonic solid left to uniquely sequence in this inward nesting of the Platonics and that is the icosahedron. Here is where the inner cube of the 3 concentric phi cubes diagrammed earlier enters into the picture.
The Greek, Philolaus (circa 470-385 B.C.E. according to Stanford Encyclopedia of Philosophy), was reputedly the earliest known Pythagorean philosopher. He produced just one written work, On Nature. Twenty plus so-called "fragments' have been attributed to Philolaus from that book but the general consensus is that only 11 of them are genuinely his while the others were forged under his name.
In fragment 12, Philolaus wrote that: "In the sphere there are five elements, those inside the
In fragment 12, Philolaus wrote that: "In the sphere there are five elements, those inside the
sphere, fire, water and earth and air, and what is the hull of the sphere, the fifth." 1 Each of the five elements is associated with one of the Platonic solids: fire the tetrahedron, water the icosahedron, earth the cube and air the octahedron. That leaves the dodecahedron which Philolaus refers to as the "hull of the sphere" and is the fifth, associated with ether (aether).
Plato (429-347 B.C.E. according to Stanford Encyclopedia of Philosophy) was allegedly heavily influenced by the Pythagoreans. It is from his name the term Platonic solids is derived. Plato celebrated these forms in his writings, most notably the Timaeus. In the Phaedo, he specifically refers to the dodecahedron in the passage: "...the real earth viewed from above is supposed to look like one of these balls made of 12 pieces of skin, variegated and marked out in different colors [i.e dodeca]."
The dodecahedron then is intimately and intricately tied in with that most perfect of 3-dimensional forms, the sphere, in the writings of these ancient Greek philosophers.
Over the years, archaeologists have uncovered evidence of the dodeca's importance during the Gallo-Roman period (1st-4th CE). About 80 small bronze "spherical" dodecahedrons have been discovered so far, primarily in Gallo-Roman sites in central France.2 They average 3" in size (about the size of a large apple) and 200 grams in weight and clearly were portable and considered precious.
The Gallo-Roman civilization was forged from the merging of two great civilizations, Gallic (Celtic) and Roman. The Celts and the Romans, along with the Greeks, were very familiar with bronze and knew it to be a stronger and more enduring alloy than iron. It was frequently used in objects considered sacred and magical. The fact that bronze was used to make these dodecahedrons indicates the objects were regarded as very precious.
Plato (429-347 B.C.E. according to Stanford Encyclopedia of Philosophy) was allegedly heavily influenced by the Pythagoreans. It is from his name the term Platonic solids is derived. Plato celebrated these forms in his writings, most notably the Timaeus. In the Phaedo, he specifically refers to the dodecahedron in the passage: "...the real earth viewed from above is supposed to look like one of these balls made of 12 pieces of skin, variegated and marked out in different colors [i.e dodeca]."
The dodecahedron then is intimately and intricately tied in with that most perfect of 3-dimensional forms, the sphere, in the writings of these ancient Greek philosophers.
Over the years, archaeologists have uncovered evidence of the dodeca's importance during the Gallo-Roman period (1st-4th CE). About 80 small bronze "spherical" dodecahedrons have been discovered so far, primarily in Gallo-Roman sites in central France.2 They average 3" in size (about the size of a large apple) and 200 grams in weight and clearly were portable and considered precious.
The Gallo-Roman civilization was forged from the merging of two great civilizations, Gallic (Celtic) and Roman. The Celts and the Romans, along with the Greeks, were very familiar with bronze and knew it to be a stronger and more enduring alloy than iron. It was frequently used in objects considered sacred and magical. The fact that bronze was used to make these dodecahedrons indicates the objects were regarded as very precious.
Above are two examples of these spherical dodecahedrons which, with the exception of one made from silver, were constructed of bronze from the Gallo-Roman times. In each case, circular openings were bored through the centers of each of the 12 pentagonal faces of the dodecas. In her book Measuring Heaven, Christiane Joost-Gaugier notes that:
Each opening is decorated with carefully incised lines forming a pentagon; thus each dodecahedron displays 12 pentagons. 3
Later on, we will see the geometry behind those 12 pentagons on the faces of the dodeca and also suggest what those circular openings on them are for. Note also the exaggerated vertices of the dodeca in the form of spherical knobs which Joost-Gaugier says appears on all the Gallo-Roman dodecahedrons found to date.
Each opening is decorated with carefully incised lines forming a pentagon; thus each dodecahedron displays 12 pentagons. 3
Later on, we will see the geometry behind those 12 pentagons on the faces of the dodeca and also suggest what those circular openings on them are for. Note also the exaggerated vertices of the dodeca in the form of spherical knobs which Joost-Gaugier says appears on all the Gallo-Roman dodecahedrons found to date.
Flatland to Sphereland & Beyond
Part II Continued
©Lorena Loo
Part II Continued
©Lorena Loo
Inward Nesting of the Platonics (Continued)
After spending a little time studying a Zometool model of an icosahedron I built, I figured out the secret to systematically constructing an icosahedron. Anybody who follows this method can easily construct an icosa from the 6 brown lines and 12 red dots on the cube seen in the last animation on the previous page.
Start with any of the red dots (endpoints of the brown lines) and connect it to its 4 nearest red dots (line endpoints) to which it is not already connected to. i.e. Each red dot is already connected to one other red dot, the one at the other end of the line it is on. So we do not include that as one of the "4 nearest neighbors. Proceed to do that with each red dot in a systematic order. At some point, you will find some of the "4 nearest neighbors" are already joined as you make your way around to all the red dots. When that happens, just join the remaining unconnected of those "4 nearest neighbors.
We end up with the icosahedron nested within the cube as the animation below illustrates. All 12 vertices of the icosa lie on the faces of the cube, 2 per face. Six of the icosa's 30 edges lie on the faces of the cube, one per face.
After spending a little time studying a Zometool model of an icosahedron I built, I figured out the secret to systematically constructing an icosahedron. Anybody who follows this method can easily construct an icosa from the 6 brown lines and 12 red dots on the cube seen in the last animation on the previous page.
Start with any of the red dots (endpoints of the brown lines) and connect it to its 4 nearest red dots (line endpoints) to which it is not already connected to. i.e. Each red dot is already connected to one other red dot, the one at the other end of the line it is on. So we do not include that as one of the "4 nearest neighbors. Proceed to do that with each red dot in a systematic order. At some point, you will find some of the "4 nearest neighbors" are already joined as you make your way around to all the red dots. When that happens, just join the remaining unconnected of those "4 nearest neighbors.
We end up with the icosahedron nested within the cube as the animation below illustrates. All 12 vertices of the icosa lie on the faces of the cube, 2 per face. Six of the icosa's 30 edges lie on the faces of the cube, one per face.
cube is also nested within the icosahedron. Again, do not be concerned about absorbing all of this at once. The combination of the words and the animation to the left will enable you understand.
So far we have looked at 4 of the 5 Platonic solids. There is still the dodecahedron but that merits a section all unto itself. In Part III, we are going to crank it up a few notches.
So far we have looked at 4 of the 5 Platonic solids. There is still the dodecahedron but that merits a section all unto itself. In Part III, we are going to crank it up a few notches.
Those 3 pairs of parallel lines on opposite faces of the cube we used to create the icosahedron also happen to define 3 golden rectangles within the icosahedron. These golden rectangles (length to width in phi to 1 ratio) are mutually orthogonal (perpendicular to one another).
The midpoints of those 6 brown lines correspond to the center of the cube faces they are located on. Remember the 6 centers of the faces of the square form the vertices of the octahedron nested within the cube. That means the octa nested within the
The midpoints of those 6 brown lines correspond to the center of the cube faces they are located on. Remember the 6 centers of the faces of the square form the vertices of the octahedron nested within the cube. That means the octa nested within the
From Milestones in the History of Polyhedra by Joseph Malkevitch
Photo Credit: Hereford Heritage Services, Hereford Museum
The dodecahedron made its appearance in human history even before it was written about by Greek philosophers or turned into mysterious bronze objects by the Gallo-Romans. Spherical forms of the dodeca as well as the other Platonic solids were carved out of stone by our human ancestors back in the Neolithic Stone Age (4,500 BCE-2,000 BCE). The Ashmolean Museum at Oxford is home to a number of these stone polyhedra which were written about by Keith Critchlow in his book Time Stands Still. To the left is the spherical stone dodeca from the Ashmolean collection.
In 2003, the spherical dodeca was proposed as
In 2003, the spherical dodeca was proposed as
From Sacred Geometry by Robert Lawlor
the model of the universe by a team of French cosmologists headed by Jean-Pierre Luminet and American mathematician Jeffrey Weeks. 4 (Image to left represents the model)
This particular shape of the universe was able to account for the data gathered from NASA's Wilkinson Microwave Anisotropy Probe (WMAP) of 2003 on the Cosmic Microwave Background. Now all scientific theories have to be taken with a grain
This particular shape of the universe was able to account for the data gathered from NASA's Wilkinson Microwave Anisotropy Probe (WMAP) of 2003 on the Cosmic Microwave Background. Now all scientific theories have to be taken with a grain
of salt for they are only hypotheses that seemingly provide the best fit to the data thus far known rather than absolute constructs of ultimate reality. In this particular case, the data gathered from the WMAP probe is interpreted in terms of Big Bang cosmology. Some prominent cosmologists maintain there never was a Big Bang and that also correlates with spirital teachings such as Vedanta, Khemitology and Buddhism. Incidentally, the late astronomer Fred Hoyle was the one who (in ridicule) coined the term "Big Bang" and he was adamantly against Big Bang cosmology.
Nonetheless it is curious how a polyhedral model that was known to our Neanderthal ancestors pops up as a model of our universe thousands of years later.
Now let's get down to the nuts and bolts of the dodecahedron. In the previous section, we saw how to construct an icosahedron from a cube. Now we are going to see how to form a dodecahedron from two cubes related by the golden ratio. First consider three successively larger cubes all having the same center. Each successive cube has edge lengths phi times that of the previous cube.
Nonetheless it is curious how a polyhedral model that was known to our Neanderthal ancestors pops up as a model of our universe thousands of years later.
Now let's get down to the nuts and bolts of the dodecahedron. In the previous section, we saw how to construct an icosahedron from a cube. Now we are going to see how to form a dodecahedron from two cubes related by the golden ratio. First consider three successively larger cubes all having the same center. Each successive cube has edge lengths phi times that of the previous cube.
In the above diagram, each edge of the outer cube is phi times the length of any edge of the middle cube. The middle cube's edges are phi times the length of any edge of the inner cube. So the outer cube's edges are phi squared (phi times phi) longer than the edges of the inner cube. Let's consider only the outer and middle cube for now. The reason why the inner cube is also illustrated will become apparent later.
On the faces of the outer cube, we do almost exactly what we did to form an icosahedron in drawing pairs of parallel lines on the opposite faces of the cube. This time, though, instead of these lines being phi times smaller in length than the edges of the cube, they are phi squared times
smaller. i.e. the length of the cube's
edges. Again the lines are centered on the faces of the cube and the red dots in the diagram to the left represent the (12) endpoints of the lines.
smaller. i.e. the length of the cube's
edges. Again the lines are centered on the faces of the cube and the red dots in the diagram to the left represent the (12) endpoints of the lines.
Each of these endpoints is exactly the same distance from two vertices of the middle cube. Think of it as the two vertices of the middle cube which are the nearest neighbors to a given (red) endpoint. There are 12 endpoints and 8 vertices on the middle cube which together serve as the 20 vertices of a dodecahedron. By joining these endpoints to its two nearest "vertice" neighbors on the middle cube, we will get a dodecahedron as the animation below illustrates.
Just like the icosahedron, we can form 3 mutually perpendicular rectangles from pairs of opposite edges. Instead of being golden rectangles like the icosa, the dodeca's rectangles have length to width ratio of phi squared to one. There are 15 such phi squared to one rectangles that can be formed from opposite pairs of edges of the dodeca.
Another way of looking at how to form a dodeca is you roof the faces of a cube. That cube just nests inside the dodeca.
Another way of looking at how to form a dodeca is you roof the faces of a cube. That cube just nests inside the dodeca.
