How do you people discover polytopes?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

How do you people discover polytopes?

Postby ubersketch » Thu Feb 01, 2018 12:51 am

I was just wondering, because I would like to discover some myself.
I assume you start with notation.
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Re: How do you people discover polytopes?

Postby quickfur » Thu Feb 01, 2018 2:32 am

ubersketch wrote:[...]
I assume you start with notation.

Nope. I start by visualizing them in my head, and then working them out on paper afterwards to make sure they actually exist. :D

On a more serious note, though: if you want to discover polytopes on your own, your first order of business is to understand the current known ones. And by that, I mean really understand them, not just go "oh yeah, I know that already, next". You should understand them to the point where if you were put into a room with nothing except a pen and paper (and no internet connection or smartphone :D), you can derive them from scratch. Well, in theory anyway. The point is that you have to thoroughly know them like the back of your hand. Or more importantly, understand how they are constructed, and the methods behind their construction. At the very least, you'll need to understand and know how to use methods like dimensional analogy, Stott expansion, and be able to compute stuff directly every now and then, e.g., compute the height of an icosahedral pyramid given only the coordinates of an icosahedron.

Yeah, it's hard work. Don't expect to be able to find something new just by dreaming it up one day waking up out of bed. Chances are, whatever you find that way has already been discovered before. :lol: (OTOH, though, being able to discover polytopes by dreaming them up at night is not a bad thing, even if what you find has already been found before. The fact that you're able to dream it up independently is a sign that you're making progress. :P Keep going, and sooner or later you'll find yourself discovering new stuff. Just don't expect that to happen overnight.)

And BTW, I'm being serious when I say that I start by visualization. This approach doesn't work for everybody, but on the off-chance it works for you, try my 4D visualization tutorial to see if it helps you get started. Of course, you won't get very far with just visualization alone; that has to be backed up by strong technical ability to work with the maths behind it. But it can be a very powerful tool to guide your way through the forest of abstract mathematical symbols that may otherwise prove too difficult to navigate. (And once you get good at it, you should be able to read the symbols and immediately visualize what they mean, not just push the symbols around without really comprehending what you're doing.)
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Re: How do you people discover polytopes?

Postby ubersketch » Thu Feb 01, 2018 11:54 pm

quickfur wrote:
ubersketch wrote:[...]
I assume you start with notation.

(snip)

How would you visualize something though? Just wondering since most of the discoveries people make are in 4D.
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Re: How do you people discover polytopes?

Postby quickfur » Fri Feb 02, 2018 12:31 am

ubersketch wrote:
quickfur wrote:
ubersketch wrote:[...]
I assume you start with notation.

(snip)

How would you visualize something though? Just wondering since most of the discoveries people make are in 4D.

Read the page linked in my previous post. It explains my method. :nod:
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Re: How do you people discover polytopes?

Postby Klitzing » Fri Feb 02, 2018 2:38 pm

ubersketch wrote:How would you visualize something though? Just wondering since most of the discoveries people make are in 4D.

Wrt. the last bit of that quote: that is wrong. It's more within this forum, that 4D gets some preference.
But even within good ol' 3D there are lots of discoveries too, cf. the French administrated, so international
Facebook page https://www.facebook.com/groups/196165767185827/.

Wrt. visualisation there has been writen an extra article onto that topic way back in 2000, giving rise to a specific
research for "easily to visualize convex polychora", then being called - and the paper was entitled accordingly -
"Convex Segmentochora". That one was published at Symmetry: Culture and Science Vol. 11, Nos. 1-4, 139-181, 2000
and could be accessed electronically online at
--- rk
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Re: How do you people discover polytopes?

Postby ubersketch » Fri Feb 02, 2018 5:59 pm

Klitzing wrote:
ubersketch wrote:How would you visualize something though? Just wondering since most of the discoveries people make are in 4D.

Wrt. the last bit of that quote: that is wrong. It's more within this forum, that 4D gets some preference.
But even within good ol' 3D there are lots of discoveries too, cf. the French administrated, so international
Facebook page https://www.facebook.com/groups/196165767185827/.

Wrt. visualisation there has been writen an extra article onto that topic way back in 2000, giving rise to a specific
research for "easily to visualize convex polychora", then being called - and the paper was entitled accordingly -
"Convex Segmentochora". That one was published at Symmetry: Culture and Science Vol. 11, Nos. 1-4, 139-181, 2000
and could be accessed electronically online at
--- rk

Oh yeah, I remember that Facebook group, I sent an invite request there. :P
But yeah, this forum does tend towards polychora (especially CRFS).
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Re: How do you people discover polytopes?

Postby quickfur » Fri Feb 02, 2018 7:39 pm

ubersketch wrote:[...]
But yeah, this forum does tend towards polychora (especially CRFS).

Not necessarily. ICN5D has been posting a whole bunch of stuff in the toratopes section, most of which are not polychora. :D

And the CRF thing was something that came about later. In the early days, the forum was not so heavy on polychora, though there was some stuff; it was mainly dealing with speculation like what would life be like if space were 4D, that sort of thing. The focus on polychora came a bit later, and only really exploded when the CRF project was started, mainly because people started (re)discovering stuff which drew more attention, and the thing just snowballed. Then CRFebruary happened, and things really began to take off, leading to the discovery of EKF polytopes and a whole bunch of stuff related to that. It used to all confined to just a single (super-huge) thread, though, until at a certain point Keiji decided to dedicate an entire forum section to the subject. That was when it sorta just took over the forum. :P

But despite all that, we used to speculate on all kinds of other (non-CRF, non-polychora) stuff too. There was a couple of big threads about different ways to structure a hypothetical 4D planet, constructing n-dimensional objects from first principles (the various rotatope/rotope/other notations), etc.. It's just that the whole idea of CRFs is so broad and relatively unexplored that it drew interest from a good number of people, including Klitzing who prior to the CRF project had already done significant research in the area that we weren't aware of at the time.

The other thing is that polychora being 4D are still relatively closer to 3D, and so are the next easiest things to work with beyond 3D, so it's a common starting point for exploring higher dimensions. There's been a bunch of stuff on the forum that goes beyond that, though. For example, even in the early days of the forum Wendy was already regularly going up to 6-10 dimensions, with her Polygloss that's specifically designed to make discussions about the specifics of these higher dimensions more precise and tractible. And CRFs themselves are also not restricted to 4D; we've dabbled in a few families of CRF constructions that extend above 4D. But the very definition of CRF means that it's heavily dependent on CRFs in lower dimensions, and since 4D CRFs aren't completely known yet, it makes it harder to prove general results in 5D and beyond. 3D CRFs have been completely enumerated already, so there's not much more to be said about them. So naturally, 4D CRFs tend to take center stage in CRF discussions.
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Re: How do you people discover polytopes?

Postby wendy » Mon Feb 05, 2018 2:01 pm

I use a great variety of different methods to make these things.

If you invent a new notation, then filling in the squares does make polytopes. But then again not. A really good notation will fill in all the blanks nicely.

Some times i visualise the things being made from the skeleton. The octagon-ball started with the six octagons following the edge of the octahedron. Then more were added, and eventually shorter struts to fill it out. It was not for some time that i realised that you could make it from a cuboctahedron, and its image rotated by 45 deg.

Other times, i start with a fragment of the surface, and walk it around. Most of the hyperbolic tilings are done this way.
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