Octahedral Ogham?
Aug. 6th, 2004 08:06 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
triadruidSo despite nearly having the "Celtic Contemplation Cube" idea finished (see previous post for updated plan), I had the good fortune to have ![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
zylch and ES over last night after Rit. Teams meeting. I was messing with a mockup of the Cube during the meeting, and was just about to start in on it again as people were leaving. These two fine folks were interested, so I showed them what I had so far. zylch posited that it might be possible to do a 'circuit' around the edges, topologically speaking. Of course we had to play with it for a while (necessitating the instruction of origami and several colored pens first) before proving to ourselves that Euler Said No; too many odd-numbered vertices. Okay, scratch that. What about going over each edge a certain number of times (like 3)? No dice (pun intended); you can circuit a cube in 17 strokes, but one side gets used 3 times, and four of them get used twice, in no graceful pattern.
The requirement to have no more than two odd-numbered vertices meant tetrahedrons, dodecahedrons, and icosahedrons were right out as well. But an Octahedron? Hmmm, that had potential... Of course that required new mockups. No origami this time, just Scotch tape and paper; we're on a schedule, man (and it's also like 10:30 PM by now)! Four edges to a vertex, means it should be possible, right? Indeed it is, and in a very exciting way:
If you can imagine this figure centered around a triple axis, then you can interlock the three 'equators' by going first around the four 'flat' edges, then turning 90 degrees and going 3/4 of the way around the 'vertical' edges, turning and completing the four sides of the 'other vertical' edges, and then completing the last side. It sounds complicated but if you've got an octahedron (like an 8-sided die) handy it'll make perfect sense. The nice thing is, all of the edges form a continuous line this way, like ogham should. But they don't have any 'associations' with the actual faces, which is annoying considering the intent of this thing.
The other alternative here is to consider each face as 'rotating' one way or the other, and interlocking them that way. Consider one of the top-half faces as rotating clockwise, ogham going around it in a circle. The next face will rotate counterclockwise, because the side it shares is going the wrong way; and so on and so forth, until you end up with four 'clockwise' faces and four 'counterclockwise' faces, like a Coriolis effect aszylch mentioned.
I'm inclining toward this method, since the faces are supposed to be thematically linked with their edges; it looks nicer to the eye, and topologically it's still a single line.
At this pointkittenpants arrives home, and decides we're all crazy. ES leaves, and the rest of us chat idly while I'm trying to come up with 8 groupings of 3 godden, as opposed to 6 groupings of 4. In theory this should make the sides more independent of each other, because you don't have to ensure that the two edges meeting at a vertex have something similar enough to the third edge of that vertex, to justify the other two faces they're about to be involved in (see Danu/Morrigan/Medb in the Cubic entry for an example). In fact, the opposite problem is true; because no edge shares more than one edge in common with each neighbor, you have to plan out in a much more long-range fashion how you're going to connect up faces. As an example for those following along at home, try connecting a side with Dagda Mor, Morrigan, and Manannan mac Lir, to a side with Dagda Mor, Lugh, and Nuada, to a side with Nuada, Brigit, and Manannan mac Lir. It doesn't work; you need one less face between Manannan and Nuada, or one more node along the chain. Now three people cannot be in any two sides for the topology to work out: this should be interesting.
In any case, the 'categorization' of the faces will probably change dramatically over time as I work out the topology problem listed above, but for now:
There may be less of a need for paper models in the future of this project. I regret not having more topology instruction when I was in school, but I'm trying to make up for it now: these are the coolest thing I have seen in a long time. They're called Schlegel graphs, and are basically a one-dimensional representation of the platonic solids. All of the nodes and arcs are right there; very neat! I need to add topology to my skill set...
Now I just have to find a source for wooden octahedrons of a reasonable size. I could do it in stone, but the level of coarseness will mean they'll have to be even bigger, and I don't have the resources to really invest in this yet. If it works, though, it may become an item for ADF Regalia...
zylch and ES over last night after Rit. Teams meeting. I was messing with a mockup of the Cube during the meeting, and was just about to start in on it again as people were leaving. These two fine folks were interested, so I showed them what I had so far. zylch posited that it might be possible to do a 'circuit' around the edges, topologically speaking. Of course we had to play with it for a while (necessitating the instruction of origami and several colored pens first) before proving to ourselves that Euler Said No; too many odd-numbered vertices. Okay, scratch that. What about going over each edge a certain number of times (like 3)? No dice (pun intended); you can circuit a cube in 17 strokes, but one side gets used 3 times, and four of them get used twice, in no graceful pattern.The requirement to have no more than two odd-numbered vertices meant tetrahedrons, dodecahedrons, and icosahedrons were right out as well. But an Octahedron? Hmmm, that had potential... Of course that required new mockups. No origami this time, just Scotch tape and paper; we're on a schedule, man (and it's also like 10:30 PM by now)! Four edges to a vertex, means it should be possible, right? Indeed it is, and in a very exciting way:
If you can imagine this figure centered around a triple axis, then you can interlock the three 'equators' by going first around the four 'flat' edges, then turning 90 degrees and going 3/4 of the way around the 'vertical' edges, turning and completing the four sides of the 'other vertical' edges, and then completing the last side. It sounds complicated but if you've got an octahedron (like an 8-sided die) handy it'll make perfect sense. The nice thing is, all of the edges form a continuous line this way, like ogham should. But they don't have any 'associations' with the actual faces, which is annoying considering the intent of this thing.
The other alternative here is to consider each face as 'rotating' one way or the other, and interlocking them that way. Consider one of the top-half faces as rotating clockwise, ogham going around it in a circle. The next face will rotate counterclockwise, because the side it shares is going the wrong way; and so on and so forth, until you end up with four 'clockwise' faces and four 'counterclockwise' faces, like a Coriolis effect as
zylch mentioned.At this point
kittenpants arrives home, and decides we're all crazy. ES leaves, and the rest of us chat idly while I'm trying to come up with 8 groupings of 3 godden, as opposed to 6 groupings of 4. In theory this should make the sides more independent of each other, because you don't have to ensure that the two edges meeting at a vertex have something similar enough to the third edge of that vertex, to justify the other two faces they're about to be involved in (see Danu/Morrigan/Medb in the Cubic entry for an example). In fact, the opposite problem is true; because no edge shares more than one edge in common with each neighbor, you have to plan out in a much more long-range fashion how you're going to connect up faces. As an example for those following along at home, try connecting a side with Dagda Mor, Morrigan, and Manannan mac Lir, to a side with Dagda Mor, Lugh, and Nuada, to a side with Nuada, Brigit, and Manannan mac Lir. It doesn't work; you need one less face between Manannan and Nuada, or one more node along the chain. Now three people cannot be in any two sides for the topology to work out: this should be interesting.In any case, the 'categorization' of the faces will probably change dramatically over time as I work out the topology problem listed above, but for now:
- Male
- Oldest
- Female
- Crafty
- Leadership
- Battle/Strength
- Out-of-your-Head
- Will
There may be less of a need for paper models in the future of this project. I regret not having more topology instruction when I was in school, but I'm trying to make up for it now: these are the coolest thing I have seen in a long time. They're called Schlegel graphs, and are basically a one-dimensional representation of the platonic solids. All of the nodes and arcs are right there; very neat! I need to add topology to my skill set...
Now I just have to find a source for wooden octahedrons of a reasonable size. I could do it in stone, but the level of coarseness will mean they'll have to be even bigger, and I don't have the resources to really invest in this yet. If it works, though, it may become an item for ADF Regalia...
