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August 19
editAre elongated bipyramids 2-isohedral?
editI want to make sure I'm understanding the "k-isohedral figures" section of Isohedral figure correctly. Is it correct that elongated bipyramids are 2-isohedral polyhedra, with the two faces within their symmetry fundamental domains being a square and a triangle? 128.237.82.113 (talk) 19:02, 19 August 2025 (UTC)
- As far as I can tell, yes. Tito Omburo (talk) 19:09, 19 August 2025 (UTC)
- The examples of elongated bipyramids given in our article are all regular, formed from a regular right bipyramid and a regular prism: the pyramids and prism are based on the same regular polygon and the apices of the pyramids are right in the middle above the centres of their bases, connected by an axis of rotational symmetry, while the sides of the prisms are squares. This regularity is sufficient for an elongated bipyramid to be a 2-isohedral figure, while the latter requires regularity of the pyramids together with the sides of the prism being rectangles – they don't have to be squares. However, the definition as given in our article of the notion of elongated bipyramid does not appear to require regularity. If you take Johnson solid J14, a 2-isohedral figure, and move its vertices just so slightly, it ceases to be 2-isohedral, but still fits our definition of elongated bipyramidality. I'm not sure what is going on; is our definition too lax or is the unwavering regularity of the examples misleading? Note that in the usual mathematical definition of pentagon it can be any five-sided polygon, while in recreational mathematical texts regularity of the polygon is invariably taken for granted. This may be a similar case. ‑‑Lambiam 08:47, 20 August 2025 (UTC)
August 23
editImpossible objects
editIs it possible to build a Penrose triangle in a 4- or 5-dimensional space? Or a big Penrose triangle in 3-dimensional space with a slow two-side torsion? And an impossible trident in a higher dimensional space?-- Carnby (talk) 10:10, 23 August 2025 (UTC)
- It is possible to have a three-dimensional figure that is topologically a torus but looks like a Penrose triangle when viewed from a particular angle. But the lighting may look off, and even if the lighting is too diffuse, it becomes clear that your eyes are being tricked as soon as it is rotated.[1][2][3] If the three bars are to be rectangular cuboids (whose intersections are cubes), it is impossible in any dimension. ‑‑Lambiam 16:31, 23 August 2025 (UTC)
- Although, as the article notes, an isometric embedding is possible in five dimensional Euclidean space. Tito Omburo (talk) 16:54, 23 August 2025 (UTC)
- It seems to me that isometry would imply that the embedded bars remain rectangular cuboids, thus contradicting my assertion above. Am I mistaken? (My intuitive insight into R5 is somewhat lacking.) ‑‑Lambiam 20:16, 23 August 2025 (UTC)
- I think there may not be a contradiction, but I haven't researched the construction in detail. In five dimensions, a rectangular box can be isometrically embedded with a bend/twist without distorting the intrinsic metric. (Similar to the isometric embedding of a flat Möbius strip in 3 dimensions or flat torus in 4-dimensions.) Tito Omburo (talk) 18:26, 26 August 2025 (UTC)
- It seems to me that isometry would imply that the embedded bars remain rectangular cuboids, thus contradicting my assertion above. Am I mistaken? (My intuitive insight into R5 is somewhat lacking.) ‑‑Lambiam 20:16, 23 August 2025 (UTC)
- Although, as the article notes, an isometric embedding is possible in five dimensional Euclidean space. Tito Omburo (talk) 16:54, 23 August 2025 (UTC)
August 26
editSources for the history of number theory
editHello. I am writing a draft at Draft:History of number theory. Id like to ask for sources that are broad in coverage. Some great ones i have found is Dickson 1952 History of the Theory of Numbers, Ore 1948 Number Theory and Its History, Watkins 2014 Number Theory: A Historical Approach. Although they are great, most are outdated. Toukouyori Mimoto (talk) 17:35, 26 August 2025 (UTC)
- P.S. general maths history books with number theory chapters are OK. Toukouyori Mimoto (talk) 17:42, 26 August 2025 (UTC)