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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)
- Wikipedia does not offer a general definition of "isometric embedding", but only one in terms of differential geometry for (pseudo-)Riemannian manifolds whose wording I do not understand. My understanding of the term "isometric embedding" was that it is a topological embedding that is also a (standard) isometry – a distance-preserving transformation. This definition would also work for metric spaces in which the notion of arclength is not meaningful. With this definition, embedded squares remain square in the metric of the target space. But I guess now this is not what the term "isometric embedding" means to differential geometers. ‑‑Lambiam 10:46, 27 August 2025 (UTC)
- It's a question of the intrinsic metric. For example, if you take a piece of paper and bend it, that's still considered an isometric embedding because distances measured intrinsically (along geodesics on the piece of paper) are unaffected. Tito Omburo (talk) 12:25, 27 August 2025 (UTC)
- Is a definition of "isometric embedding" in terms of "intrinsic metric" also equivalent to the present definition for pseudo-Riemannian geometry? If so, I think Wikipedia should present this alternative definition first before giving the present one (which is, essentially, that the source metric is the pullback of the target metric). ‑‑Lambiam 09:12, 28 August 2025 (UTC)
- Incidentally, there are good reasons to be skeptical that this is the minimum dimension. I suspect there may be a isometric embedding in four dimensional Euclidean space, analogously to the embedding of a flat torus in three dimensions. Tito Omburo (talk) 12:36, 27 August 2025 (UTC)
- It's a question of the intrinsic metric. For example, if you take a piece of paper and bend it, that's still considered an isometric embedding because distances measured intrinsically (along geodesics on the piece of paper) are unaffected. Tito Omburo (talk) 12:25, 27 August 2025 (UTC)
- Wikipedia does not offer a general definition of "isometric embedding", but only one in terms of differential geometry for (pseudo-)Riemannian manifolds whose wording I do not understand. My understanding of the term "isometric embedding" was that it is a topological embedding that is also a (standard) isometry – a distance-preserving transformation. This definition would also work for metric spaces in which the notion of arclength is not meaningful. With this definition, embedded squares remain square in the metric of the target space. But I guess now this is not what the term "isometric embedding" means to differential geometers. ‑‑Lambiam 10:46, 27 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)
- Number Theory and Geometry through History (2025).
- Trilogy Of Numbers And Arithmetic — Book 1: History Of Numbers And Arithmetic: An Information Perspective (2022) (accessible online).
- History of Mathematics, volume II: Special topics of elementary mathematics (1925), downloadable here, has a brief section (pp. 4–6) on "Early Writers on Number Theory" and further historical discussion of specific number-theoretical topics sprinkled throughout the book.
- ‑‑Lambiam 11:29, 27 August 2025 (UTC)
- There is Weil's "Number Theory, an approach through history", but it is still as dated as these sources. Rademacher wrote a very good mid-century paper on the history of analytic number theory: Hans Rademacher, “Trends in research: The analytic number theory,” Bulletin of the American Mathematical Society 48 (6), 379–401 (June 1942). Tito Omburo (talk) 11:46, 27 August 2025 (UTC)
- Your current draft omits development of algebraic number theory. Here are some lecture notes, all of which I believe is standard and can be given better sources if necessary. Tito Omburo (talk) 22:46, 27 August 2025 (UTC)
- I've given a partial list of things missing on the talk page of the draft. Tito Omburo (talk) 11:21, 29 August 2025 (UTC)
- Thank you, all! Toukouyori Mimoto (talk) 17:01, 29 August 2025 (UTC)
August 27
editTerm for a spiral in a square
edit
Does the shape (fractal) in http://numerade.com/ask/question/python-turtle-draw-the-following-striped-square-spiral-modify-the-code-of-previous-problem-58538 have a specific name in mathematics? "Square spiral" seems to refer to one from which the Ulam spiral is constructed.
Thanks, cmɢʟee⎆τaʟκ 19:54, 27 August 2025 (UTC)
- This appears to be a mutual pursuit curve starting at the four corners of a square. You can find more information and images at [4], with the special case of a square starting about half way down (search for "case of a square"). A single branch turns out to be a logarithmic spiral. RDBury (talk) 21:39, 27 August 2025 (UTC)
- Thanks, RDBury. Good to know, cmɢʟee⎆τaʟκ 21:45, 27 August 2025 (UTC)
- See Mice problem. ‑‑Lambiam 09:46, 28 August 2025 (UTC)
calculating length of mountain-shadow on clouds, and from how far away it's visible
edit
If the sun is near the horizon and there are clouds above a mountain (but not blocking the sun), the sun may illuminate clouds except where blocked by the mountain: the mountain casts a shadow onto the clouds, like in the image. The clouds can be higher than the mountain, and the shadow extends away from the mountain, so I assume the shadow is visible from further away than the mountain itself is visible from. I'm trying to figure out what formula I would use to calculate how far away from a mountain (of height M) a shadow (on clouds of height C) - and the illumination / non-shadow to either side of it - extends. Does anyone know how to calculate this? Second question, if I want to then calculate how far away from that shadow a person on the ground at sea level can be and still see it, is the formula for that basically just this aka this (measuring from the height of the clouds to an observer at sea level)? Buildingquestion (talk) 22:15, 27 August 2025 (UTC)
- Here is a link to an earlier similar question: Wikipedia:Reference desk/Archives/Mathematics/2025 July 3 § Geometry / atmospherics query.
- A formula for the distance to the horizon is given at Horizon § Formula. This is essentially an application of the "Power of a point" formula; see the first figure in that article.
- Does this contain enough material for you to find the answers to your present questions? ‑‑Lambiam 09:41, 28 August 2025 (UTC)
- I think it answers the second question, but it is not obvious to me that it answers the first question. Are you suggesting that if the distance to the horizon from the height of the mountain (M) is X, and the distance to the horizon from the height of the clouds (C) is Y, then the difference between X and Y is also the distance that the shadow falling on the clouds extends away from the mountain? Buildingquestion (talk) 20:03, 29 August 2025 (UTC)
- No. I hoped the mathematical model given in my answer to the earlier question might guide your thinking. The problem as stated is in fact not well defined. Assume an observer (possibly a drone with a camera) flying in the shadow of the mountain. It cannot see any part of the solar disk. Now if it moves out in a straight line, parallel to the solar rays, it will initially remain in that shadow. If the drone flies into a cloud, that cloud will be (possibly partially) in the shadow of the mountain if the drone was at that time in its shadow.
- But as the observer moves farther and farther out, the angular diameter of the mountain shrinks inversely proportional to the distance, whereas that of the solar disk remains virtually constant. So at some point the mountain is an insignificant speck against the solar disk – of which only part may be visible to the observer, but that is then because it is blocked by the horizon, not the mountain per se. At what distance do we decide the drone is no longer in the mountain's shadow? When not all of the solar disk is blocked from view (penumbral)? But that does not only depend on the height of the mountain, but also on its shape; for a needle-shaped mountain it will be much earlier than for a dome shape.
- The drone can also fly away at an angle, in a straight line from where it is to (just above) the horizon, continuing from there in a straight line. Then it is more likely it remains in the shadow all the way. The limit might then become how far away a cloud can be before all clouds are below the horizon, which is determined by the maximum height at which there can be clouds.
- Apart from the above, the unspecified angle between the direction of the Sun and the vertical at the mountain top can make a difference. ‑‑Lambiam 22:09, 29 August 2025 (UTC)
- I think it answers the second question, but it is not obvious to me that it answers the first question. Are you suggesting that if the distance to the horizon from the height of the mountain (M) is X, and the distance to the horizon from the height of the clouds (C) is Y, then the difference between X and Y is also the distance that the shadow falling on the clouds extends away from the mountain? Buildingquestion (talk) 20:03, 29 August 2025 (UTC)
August 28
editPrime number theorem
editWhilst reading "Sources for the history of number theory" above, I wandered my way to Prime number theorem. Reading the introductory paragraph and the Statement section, pi(x) ~= x/ln(x) makes sense. Then I get confused. What does Li stand for (and should it be "li"?)? Where does pi(x) = integral[2 to x](1/ln(t))dt come from? It just seems to spring from midair. -- SGBailey (talk) 13:11, 28 August 2025 (UTC)
- Tito Omburo (talk) 13:28, 28 August 2025 (UTC)
- This should obviously not be sprung like that, without warning, excuse or explanation, upon the unsuspecting reader. A definition of is given in the section Prime number theorem § Prime-counting function in terms of the logarithmic integral, which should also explain how approximates ‑‑Lambiam 16:38, 28 August 2025 (UTC)
