Questions about Stellations and Stellation Criterion
Posted: Sat Jul 16, 2011 2:27 pm
This thought started a few days ago when I thought I'd try and export images of each stellation of the icosidodecahedron. I noticed there wasn't anything out there listing each one, as there is for the platonic solids, so I thought I'd give it a go (and save a few nets as PDF's along the way so I could build them in the future)
There were far more than I expected using the 'fully supported' stellation mode. I quickly googled for the stellations and came across this site:
http://mathworld.wolfram.com/Icosidodec ... tions.html
Where it says there are 432 enantiomorphous and 415 chiral stelations (I presume 847 in total). However, using Miller's rules gives at least 7million.
Obviously, although making images of all 847 fully supported stellations is doable, and I'm already over half way, I don't think I'd attempt to view all those Miller criterion generated ones...
Then I thought, while I'm playing around with these stelation modes, why not have a look at the stelations for a number of the snubs. My attention quickly turned to each one in turn, until I came to the Great Dirhombicosidodecahedron.
I set the stelation criterion to mainline, viewing the polyhedron in the left window, and the evolving stelation diagram in the right. This way I was able to get a 'feel' for those higher stellations that otherwise would have taken forever to reach using the fully supported criterion.
Then, something strange happened. As I incremented up each stelation in turn, the polyhedron eventually turned into a sort of truncated icosidodecahedron. I continued, but ended up in a continuous loop which displays the stelations of this truncated model instead of the original snub.
Next, I thought I'd view the stelations of the Small Inverted Retrosnub Icosicosidodecahedron. I used spheres and cylinders in the display, set to 'inverse square', with a sphere radios of 0.23 and cylinder radius ratio of 0.49.
Moving through the stelations proved no issue until I suddenly noticed the spheres were getting larger with each step, until they filled the entire volume of the polyhedron. Clicking the stelation down arrow to cycle downwards gave the same result (and much quicker at reaching it too). Again, I was caught in a strange loop where I could not get back to the original polyhedron unless I selected "original polyhedron" in the stellation menu. Again, as with stellating Miller's monster, I had the stellation mode set to 'mainline'.
Any ideas why these anomalies happen? Is it a bug, or something to do with the way the stelations are generated mathematically.
Twi
There were far more than I expected using the 'fully supported' stellation mode. I quickly googled for the stellations and came across this site:
http://mathworld.wolfram.com/Icosidodec ... tions.html
Where it says there are 432 enantiomorphous and 415 chiral stelations (I presume 847 in total). However, using Miller's rules gives at least 7million.
Obviously, although making images of all 847 fully supported stellations is doable, and I'm already over half way, I don't think I'd attempt to view all those Miller criterion generated ones...
Then I thought, while I'm playing around with these stelation modes, why not have a look at the stelations for a number of the snubs. My attention quickly turned to each one in turn, until I came to the Great Dirhombicosidodecahedron.
I set the stelation criterion to mainline, viewing the polyhedron in the left window, and the evolving stelation diagram in the right. This way I was able to get a 'feel' for those higher stellations that otherwise would have taken forever to reach using the fully supported criterion.
Then, something strange happened. As I incremented up each stelation in turn, the polyhedron eventually turned into a sort of truncated icosidodecahedron. I continued, but ended up in a continuous loop which displays the stelations of this truncated model instead of the original snub.
Next, I thought I'd view the stelations of the Small Inverted Retrosnub Icosicosidodecahedron. I used spheres and cylinders in the display, set to 'inverse square', with a sphere radios of 0.23 and cylinder radius ratio of 0.49.
Moving through the stelations proved no issue until I suddenly noticed the spheres were getting larger with each step, until they filled the entire volume of the polyhedron. Clicking the stelation down arrow to cycle downwards gave the same result (and much quicker at reaching it too). Again, I was caught in a strange loop where I could not get back to the original polyhedron unless I selected "original polyhedron" in the stellation menu. Again, as with stellating Miller's monster, I had the stellation mode set to 'mainline'.
Any ideas why these anomalies happen? Is it a bug, or something to do with the way the stelations are generated mathematically.
Twi