# User talk:Sycamore916

Hi, are you on the Polytope Discord? It's where we discuss the organisation of the wiki (among other polytope related things). As an active editor, it would be nice to have you there. --Galoomba (talk) 08:22, 28 December 2022 (UTC)

- Hi! I'm not on the polytope discord. You can reach me here though. Sycamore916 (talk) 14:09, 28 December 2022 (UTC)

## Stewart toroids[edit source]

Hi! Making pages for Stewart toroids has been a goal of mine for a while, and I'm glad someone finally did it. But there are some things I'd like your opinion on before I write/change them, since it's clear you've read Adventures much more closely than I have.

- I'm pretty sure there's more to the (T) requirement than you describe on the Stewart toroid page. The text I'm used to here says every polyhedral "unit" of the toroid, be it augmentation or excavation, has to change the genus, it can't be a superfluous addition.
- How much do you want to add about knotted toroids? One can do far more than just stretch Chapter-7-like loops in the way Chapter 18 does. I've found several elegant knotted toroids with H
_{3}+ symmetry (some using m, in case that interests you), and I think Albert P. Carpenter has made a few knotted toroids as well. - Do you think we can fit the Stewart notation into a section on the page Nomenclature? Or does it really need its own page?
- And I want to make sure my use of the Stewart notation is passable before I start writing it all over the place. I only read Adventures a few times and then went down a rabbit hole where I needed to adapt the notation to Chapter-7-like toroids with many different units. The excavated truncated rhombicuboctahedron I would write as 6Q
_{4}8(Q_{3}P_{3}^{3A}), or maybe 6Q_{4}8Q_{3}24P_{3}. How would you go about expressing this or the Webb toroid in Stewart notation? - Also, I intend to put images on your page Quasi-convexity eventually. Even if the concept is relatively simple to understand, I like to think the pictures help people get it.
- Also, someone found two new quasi-convex J76-based toroids here

I'm very excited to start working with you. The New Kid (talk) 11:37, 3 April 2023 (UTC)

- Hi! It would be nice to have another person contributing on the Stewart toroids. I've taken a bit of a break from them, but there's tons I'd still like to add. I've browsed your TODO page quite a bit over the course of things.
- Yes I think the (T) requirement is more complex. I've changed my mind on what exactly (T) means a few times. At an earlier point I thought Stewart was using round-about phrasing to essentially say that the genus was positive, but I acknowledge now that there is some subtlety. My current understanding of (T) is, as the linked page says, it is an extremely weak condition. A strict reading of Stewart's wording would have (T) rule out not even the platonic solids, although I think Stewart intended for it to rule out genus 0 polytopes. If you consider (R)(A)(Q) polyhedra with positive genus the number of polyhedra ruled out by (T) is an extremely small fraction. At the moment I think one of the following is true of (T) and explains its weirdness:
- (T) is a condition Stewart invented at a point where the number of (R)(A)(Q) polyhedra was believed to be much smaller. Now we know that there are sextillions of (R)(A)(Q)(T) polyhedra, but it seems initially Stewart believed that there were a few hundred. It's possible (T) used to rule out a larger percentage of the known shapes and it has just stuck around past the point of it's usefulness.
- Stewart intended something different by (T) than how I am understanding it. The wording of (T), and the common understanding, seem to suggest that this is some sort of minimality condition, meant to avoid extraneous excavations. However it doesn't really succeed at that. It's possible Stewart had the ill-defined idea of "no extra excavations" and thought that the wording of (T) ruled out extra excavations to an extent far greater than it actually does.

- Either way I am left unsure about (T). Because I'm unsure I haven't updated the definition I wrote earlier. I think it would be best probably to copy the definition verbatim from the book until we can come to some closure on what exactly it means. If you or anyone else thinks you understand (T) fully, I'd love to hear about it, feel free to edit.

- I don't find knotted toroids too interesting. I think they are what a mentor of mine would call a "money pit". You can do pretty much endless research into them and continue to come up with novel results indefinitely they are just unlikely to be interesting. Without stronger properties to narrow them down or classify them they are just too diverse of a group. I'd like to copy what's in the book, but I don't personally plan to go much farther than that. I'm absolutely not going to stop others in any way. A bit of a weak statement, but if there's anything interesting in that realm I think it would be cool to add it.
- I think the Stewart notation should have its own page. Although I might not be the best to ask since I don't think glossary or compilations pages are very useful in the first place. I just find it a little hard to imagine someone who just wants to learn about any notation at all, which seems like the target for such pages. I find it more likely someone wants to understand a specific piece of notation or a specific term. Stewart's notation is pretty complex so there's a lot to say.
- As far as notation for the Webb toroid goes, my feeling with Stewart notation is that they are a bit like OBSAs. There's certainly structure there, but if Stewart didn't specifically refer to a particular polyhedron at some point, then it doesn't really have Stewart notation. OBSAs are definitely a more extreme example, Stewart notation has a stronger structure and more firm rules, certainly, but the Webb toroid is pretty different from Stewart's (R)(A)(Q)(T) polyhedra so it's a bit hard to say.
- Pictures are nice, every article that can have a useful picture definitely should. I tried to come up with some useful pictures for quasi-convexity, but I found it hard to illustrate, so have at it, I look forward to some nice pictures.
- Those are neat. Have you seen User:Sycamore916/Sandbox/Tunnellings_of_Johnson_solids? It's very much a WIP, but I've been collecting some stuff like that there.

- Yes I think the (T) requirement is more complex. I've changed my mind on what exactly (T) means a few times. At an earlier point I thought Stewart was using round-about phrasing to essentially say that the genus was positive, but I acknowledge now that there is some subtlety. My current understanding of (T) is, as the linked page says, it is an extremely weak condition. A strict reading of Stewart's wording would have (T) rule out not even the platonic solids, although I think Stewart intended for it to rule out genus 0 polytopes. If you consider (R)(A)(Q) polyhedra with positive genus the number of polyhedra ruled out by (T) is an extremely small fraction. At the moment I think one of the following is true of (T) and explains its weirdness:
- Looking forward to expanding the coverage of Stewart toroids. Sycamore916 (talk) 23:42, 4 April 2023 (UTC)
- I've studied your TODO and Sandbox pages for a while as well. Relevant to the former, I've had a 3D model of the Holey Monster for years now, and it won't be too difficult to give it a good coloring and take some pictures of the different layers.
- I don't have a full understanding of (T). My view was pretty much the same as the "common understanding" laid out in your second option there. I haven't thought about it enough, but I think copying the book's definition would be a good idea.
- Fair enough. What information is even in the book regarding knotted toroids? Probably a claim that any knot can be given the form of a regular-faced polyhedron? I don't think we have any interesting results on them (at the moment), not counting their looks.
- That makes sense. Giving the notation its own page will give it room to breathe. And it's probably better to avoid keeping all the notations on one giant page.
- That's alright. In any situation where I'd want to use an "unofficial Stewart notation," it'd probably be clearer to just use words to describe the construction.
- What ideas did you have for a picture to represent quasi-convexity? My first thought is a recognizable solid with a relatively simple tunnel through two of its faces. It should be clear that the object isn't convex, even though its convex hull is unaltered.
- I have seen this page, I will try to complete more of it eventually (unless you don't want that and the edits by HAM were unwanted). The New Kid (talk) 11:03, 9 April 2023 (UTC)

- I've studied your TODO and Sandbox pages for a while as well. Relevant to the former, I've had a 3D model of the Holey Monster for years now, and it won't be too difficult to give it a good coloring and take some pictures of the different layers.

## Hendecaxennon[edit source]

Please change the name of the hendecaxennon to hendecaronnon because ronna- is now the official prefix. Likewise the next member should be dodecaquetton. 99.157.99.148 01:18, 2 August 2023 (UTC)

- "Official"? According to whom? If it were up to me the page would be called 10-simplex, but I don't control the names. Sycamore916 (talk) 02:08, 2 August 2023 (UTC)