© 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
Golden Secret of the Dodeca (Continued)
Now let's focus on just two opposite edges on the same face of the cube as one of the edges is ratcheted through successive 72º rotations about the center of a pentagonal face.
Now let's focus on just two opposite edges on the same face of the cube as one of the edges is ratcheted through successive 72º rotations about the center of a pentagonal face.
The above nested Platonics were created with the wonderful open source virtual Zome software ZomeCAD by Elizabeth Schlapp. ZomeCAD is probably easier to learn than vZome but still requires a certain familiarity with rZome (real Zome) and some knowledge of symmetry. Thanks Elizabeth for ZomeCAD.
The renown yogi Paramahansa Yogananda said that the five elements which comprise the entire universe also makeup the human body. The rays of these 5 elements are represented by the starlike shape of the human body (see Agrippa's pentagram man diagram in previous section) with the head, feet and hands forming the 5 points of the star. Yogananda further states:
The five fingers also represent the five five vibratory elements of the Cosmic Intelligent Vibration that maintain the structure of creation. The thumb represents the grossest vibratory element, earth; hence its thickness. The first finger represents the water element. The second finger represents the darting fire element; that is why it is the longest. The third finger represents air. The smallest finger represents ether, which is very fine.5
Here finally are a couple of photos of real Zome models of the 5 unit nested Platonic solids dubbed Kepler's Obsession by Zome. Thanks to Pam White at Zome who got my parts order to me last month before Christmas. Manually advance forward or backward to get the two different views.
The renown yogi Paramahansa Yogananda said that the five elements which comprise the entire universe also makeup the human body. The rays of these 5 elements are represented by the starlike shape of the human body (see Agrippa's pentagram man diagram in previous section) with the head, feet and hands forming the 5 points of the star. Yogananda further states:
The five fingers also represent the five five vibratory elements of the Cosmic Intelligent Vibration that maintain the structure of creation. The thumb represents the grossest vibratory element, earth; hence its thickness. The first finger represents the water element. The second finger represents the darting fire element; that is why it is the longest. The third finger represents air. The smallest finger represents ether, which is very fine.5
Here finally are a couple of photos of real Zome models of the 5 unit nested Platonic solids dubbed Kepler's Obsession by Zome. Thanks to Pam White at Zome who got my parts order to me last month before Christmas. Manually advance forward or backward to get the two different views.
See how another regular pentagon is formed where the vertices are all vertices of the dodeca? This pentagon has sides of length phi times that of the dodeca's edge lengths. There are 12 such pentagons formed from the ratcheting of the cube (i.e. the five cubes nested in the dodeca). The 5 sides of each pentagon is formed by an edges of the 5 cubes, one from each. The 72º angle we ratchet successively through is also intimately connected with phi. It happens to be two of the angles of the golden triangle where the side to base ratio is phi. Five and phi-ness and phi-ness means the square root of 5. There is still much more.
Where there is a nested cube in the dodeca, there is a nested tetra. So not only are there 5 unique cubes in the dodeca, there are also 5 unique tetrahedrons, each corresponding to a unique orientation of one of the 5 cubes nested in the dodeca. But an octa is nested in a tetra nested in a cube so each of the 5 cubes and 5 tetras are associated with a unique orientation of the octahedron with respect to the dodeca.
The same does not hold true for the icosahedron within this unique nesting of Platonic solids. This is because of the fivefold symmetry of the icosa and its orientation with respect to the dodeca in this nested set. After every 72º rotation, the icosa will be oriented in exactly the same way with respect to the dodeca as it was before the rotation. i.e. The icosa will look as if it has not rotated at all unlike the other three Platonic solids.
The slideshow below illustrates some of the various configurations just mentioned. There are 6 images set on manual advance for your viewing leisure. Click on the arrows to advance forward or backward in the image set.
Where there is a nested cube in the dodeca, there is a nested tetra. So not only are there 5 unique cubes in the dodeca, there are also 5 unique tetrahedrons, each corresponding to a unique orientation of one of the 5 cubes nested in the dodeca. But an octa is nested in a tetra nested in a cube so each of the 5 cubes and 5 tetras are associated with a unique orientation of the octahedron with respect to the dodeca.
The same does not hold true for the icosahedron within this unique nesting of Platonic solids. This is because of the fivefold symmetry of the icosa and its orientation with respect to the dodeca in this nested set. After every 72º rotation, the icosa will be oriented in exactly the same way with respect to the dodeca as it was before the rotation. i.e. The icosa will look as if it has not rotated at all unlike the other three Platonic solids.
The slideshow below illustrates some of the various configurations just mentioned. There are 6 images set on manual advance for your viewing leisure. Click on the arrows to advance forward or backward in the image set.
To Be Continued
After some thought, I decided I would dedicate a few paragraphs to the Buddhist cosmology which relates to the five elements. This will be part of the next instalment which will also include mention of the Mayan timestar and geometries.
Included below are the references, footnotes and image credits thus far for Part III.
Due to some health issues, I have spent a lot of time off the computer both out of necessity and choice for a number of weeks. The time was utilized to deepen my spiritual practice.
As I have found it necessary to clarify some questions I have in my own mind regarding facets of Buddhist cosmology, I am seeking some answers first before continuing with the next section.
I have completed the move to the new host server for this site and believe the various technical hiccups have been cleared up.
Thank you for your patience.
Footnotes:
1 Der Fragmente der Vorsokratiker, Vol 1 (1951), 6th edition; Hermann Diels; pg 412f. See also Hamlet's Mill: An Essay On Myth and the Frame of Time; Giorrgio de Santillana and Hertha von Dechend; pg 232; 1969, Gambit Inc. Boston. The latter citation gives the English translation.
2 Depending on the source consulted, the number of these (bronze) spherical dodeca found from Gallo-Roman times range from 50 to 100.
3 Measuring Heaven: Pythagoras And His Influence on Thought And Art in Antiquity And the Middle Ages; Christian Joost-Gaugier; pg 205; Cornell University press, 2006.
Note: David J. Stevens, Social History Curator of the Hereford Heritage Museum informed me that the Hereford bronze dodeca does not have the incised lines around the openings.
4 Nature, Vol 425, October 9, 2003; pg 593-595; PDF version available here. Luminet's web page.
5 How You Can Talk With God; Paramahansa Yogananda; pg 19; Self-Realization Fellowship; 11 printing; 1990
Images:
Hereford Museum Bronze Dodecahedron: Special thanks to the museum for granting me permission to use the photo. Special thanks to the museum's Social History Curator, David J. Stevens, for his help in providing me detailed information on the dodeca's measurements.
(Image credits from book sources will be posted shortly.)
Included below are the references, footnotes and image credits thus far for Part III.
Due to some health issues, I have spent a lot of time off the computer both out of necessity and choice for a number of weeks. The time was utilized to deepen my spiritual practice.
As I have found it necessary to clarify some questions I have in my own mind regarding facets of Buddhist cosmology, I am seeking some answers first before continuing with the next section.
I have completed the move to the new host server for this site and believe the various technical hiccups have been cleared up.
Thank you for your patience.
Footnotes:
1 Der Fragmente der Vorsokratiker, Vol 1 (1951), 6th edition; Hermann Diels; pg 412f. See also Hamlet's Mill: An Essay On Myth and the Frame of Time; Giorrgio de Santillana and Hertha von Dechend; pg 232; 1969, Gambit Inc. Boston. The latter citation gives the English translation.
2 Depending on the source consulted, the number of these (bronze) spherical dodeca found from Gallo-Roman times range from 50 to 100.
3 Measuring Heaven: Pythagoras And His Influence on Thought And Art in Antiquity And the Middle Ages; Christian Joost-Gaugier; pg 205; Cornell University press, 2006.
Note: David J. Stevens, Social History Curator of the Hereford Heritage Museum informed me that the Hereford bronze dodeca does not have the incised lines around the openings.
4 Nature, Vol 425, October 9, 2003; pg 593-595; PDF version available here. Luminet's web page.
5 How You Can Talk With God; Paramahansa Yogananda; pg 19; Self-Realization Fellowship; 11 printing; 1990
Images:
Hereford Museum Bronze Dodecahedron: Special thanks to the museum for granting me permission to use the photo. Special thanks to the museum's Social History Curator, David J. Stevens, for his help in providing me detailed information on the dodeca's measurements.
(Image credits from book sources will be posted shortly.)