https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions tagged with Mathematics sorted by active. https://community.wolfram.com/groups/-/m/t/3695977 Find the parametric equations of the negative-pedal surface of the elliptic cylinder: {3 Cos[u], 2 Sin[u], v} with respect to the origin. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/501cac52-398a-421b-b27f-853b3054993a Alejandro Latorre Chirot 2026-04-18T20:07:43Z https://community.wolfram.com/groups/-/m/t/3695968 In this notebook we will see the third property of the points of Taylor's circumference, the three circles that each pass through the vertex of the triangle and through the foot of each perpendicular to each of the feet of the heights that are on the sides that form the same vertex. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/c1a96910-2d04-40e6-9c82-475fbb34249f Alejandro Latorre Chirot 2026-04-18T18:53:27Z https://community.wolfram.com/groups/-/m/t/3696227 $\left[ \left( (i!)^{\cos(x!) + \sin(i!)} \right)! \right]^i = \phi^{\phi!}$ Dagaga Addisu 2026-04-18T16:59:48Z https://community.wolfram.com/groups/-/m/t/3694198 ![All elementary functions from a single binary operator][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Allelementaryfunctionsfromasinglebinaryoperator.png&userId=20103 [2]: https://www.wolframcloud.com/obj/f9491e4a-5414-4fce-89f3-f455dbd662a1 Andrzej Odrzywolek 2026-04-17T21:05:07Z https://community.wolfram.com/groups/-/m/t/3694082 Find the parametric equations of the pedal and negative-pedal surfaces of the Plücker conoid: {0, 0, Sin[2 u]} + v {Cos[u], Sin[u], 0} with respect to the origin. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/8dfbede9-40c6-4e29-9457-14e8e14497c9 Alejandro Latorre Chirot 2026-04-17T15:29:15Z https://community.wolfram.com/groups/-/m/t/3694145 In this notebook we will see the second property of the points of Taylor's circumference, the three circles that pass each by two feet of the heights of two sides that form a vertex of the triangle and by the feet of two of the perpendicular to these two feet of the heigts that are on the sides that form the same vertex. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/f36bc52d-7584-43e6-9696-47089e8e099c Alejandro Latorre Chirot 2026-04-17T14:00:41Z https://community.wolfram.com/groups/-/m/t/3690608 I came across this paper (https://arxiv.org/pdf/2603.21852) published this month by Andrzej Odrzywolek and have not been able to stop thinking about it. The claim is that a single binary operator, $\text{eml}(x, y) = \exp(x) - \ln(y)$, paired with the constant 1, generates every elementary function. NAND gates do this for Boolean logic, but continuous math has never had an equivalent, and the standard reduction via Euler and Liouville bottoms out at exp, log, and arithmetic. Odrzywołek shows that the bottom is lower. So $e^x = \text{eml}(x, 1)$, $\ln x = \text{eml}(1, \text{eml}(\text{eml}(1, x), 1))$, and every elementary formula becomes a binary tree over the grammar $S \to 1 \mid \text{eml}(S, S)$. Alisson Silva 2026-04-16T11:23:02Z https://community.wolfram.com/groups/-/m/t/3690751 ![Mathematical Games: space groups and filling space][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=SpaceGroupsandFillingSpace.png&userId=20103 [2]: https://www.wolframcloud.com/obj/fb4a4c54-b4d7-4f9b-a93a-21427316f9c5 Ed Pegg 2026-04-16T15:57:25Z https://community.wolfram.com/groups/-/m/t/3690624 Taylor's circumference is a particular case of Tucker's circles. Taylor's circumference is that passes through the feet of the perpendiculars drawn from the feet of the heights of the sides of the triangle. In this notebook we will see the first property of the points of Taylor's circumference, the three circles that pass through each vertex of the triangle and by the foot of a height on the opposite side to the vertex and by the feet of the perpendiculars to that foot of the height on the other two sides of the triangle. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/116f7210-d949-476e-bcc9-f9ad2d752392 Alejandro Latorre Chirot 2026-04-16T14:01:27Z https://community.wolfram.com/groups/-/m/t/3688956 I have two vectors, v1 and v2. I can plot them and calculate the angle between them without any problem. I would like to plot this angle (theta) between them on my plot. I know that I could draw an arc between the two vectors with a small radius to show an angle measurement, but I am unsure how I can navigate the new plane defined by the two vectors. Any suggestions as to how I could do this? My work is attached. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/13054823-0185-4140-84b6-39167a6c4f2d Patrick McMullen 2026-04-15T21:45:09Z https://community.wolfram.com/groups/-/m/t/3690586 Hello Wolfram Community, I want to share a computational observation that I have not seen discussed in this precise form, and I would genuinely welcome input on whether there is an obvious derivation I am missing. ## The observation Consider the coefficient: Gcurvature[d_] := Gamma[1 + d]^2 / Gamma[1 + 2 d] For integer arguments, this equals `1/Binomial[2n, n]` — the reciprocal of the central binomial coefficients. The first few values: Gcurvature(1) = 1/2, Gcurvature(2) = 1/6, Gcurvature(3) = 1/20, Gcurvature(4) = 1/70 I have observed that the critical exponents of the three-dimensional Ising universality class appear to be expressible as exact rationals built from these four coefficients. ## The three main expressions eta3D = Gcurvature[3]^3 / (Gcurvature[1]^3 * Gcurvature[2]^2) = 9/250 = 0.0360 nu3D = Gcurvature[3]^3 / (Gcurvature[1] * Gcurvature[2]^2 * Gcurvature[4]) = 63/100 = 0.6300 beta3D = Gcurvature[4]^2 / (Gcurvature[1]^2 * Gcurvature[3]^2) = 16/49 = 0.326531... The numerator of `nu3D` simplifies remarkably: `5040/8000 = 7!/20^3`. ## Comparison with the conformal bootstrap The state-of-the-art values from Chester et al. (2024, arXiv:2411.15300) are: eta = 0.0362976(5) — GOD prediction 9/250 — error 0.82% nu = 0.6299710(4) — GOD prediction 63/100 — error 0.005% beta = 0.3264187(6) — GOD prediction 16/49 — error 0.034% By standard scaling relations, the remaining exponents also become exact rationals: gamma = 30933/25000 (bootstrap: 1.237076, error 0.020%) alpha = 11/100 (bootstrap: 0.110087, error 0.079%) delta = 1241/259 (bootstrap: 4.789843, error 0.035%) ## The 2D case is exact For the 2D Ising model (Onsager 1944), the same framework gives exact results: eta_2D = Gcurvature[1]^2 = 1/4 (Onsager: 1/4) EXACT nu_2D = Gcurvature[0] = 1 (Onsager: 1) EXACT beta_2D = Gcurvature[1]^3 = 1/8 (Onsager: 1/8) EXACT Mean-field values (d >= 4) also match exactly: eta_MF = 0 EXACT nu_MF = Gcurvature[1] = 1/2 EXACT beta_MF = Gcurvature[1] = 1/2 EXACT ## Context This observation forms part of a broader framework I have been developing (GOD Theory, DOI: 10.5281/zenodo.19599917), in which these coefficients arise as curvature coefficients of a fractal algebra built on fractional derivatives. However, the numerical match above holds independently of that framework and can be verified by anyone with the attached notebook. ## What I am asking The 2D Ising match and the mean-field match are exact. The 3D match is empirical at the 0.005%-0.82% level. I do not have a rigorous derivation of the exponent patterns (why specifically `Gcurvature[3]^3 / (Gcurvature[1]^3 * Gcurvature[2]^2)` for eta, for example). My question to the Community: does anyone see an obvious path to this via the renormalisation group, conformal field theory, or the structure of central binomial coefficients? Has any similar pattern been observed? The attached notebook contains all computations in roughly 30 lines of Wolfram Language. Please feel free to run it, criticize it, or extend it. Thanks in advance. Francisco Torrado Cano Independent Researcher Cáceres, Spain Francisco Torrado Cano 2026-04-16T11:00:14Z https://community.wolfram.com/groups/-/m/t/3690123 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/f7fbd8d8-56c6-4e84-9db6-cbab093d58c5 Theo Vine 2026-04-16T07:15:42Z https://community.wolfram.com/groups/-/m/t/3682367 ![enter image description here][1] For options A, B, C, and D, how to draw visual diagrams for all valid path scenarios that meet the conditions? For each distinct case, draw a detailed diagram, and use these visual illustrations to help understand and solve the problem. (*Find all valid non-negative integer solutions*)sol = FindInstance[{x + y + z + w == 4, x - y == -2, w == z, x >= 0, y >= 0, z >= 0, w >= 0}, {x, y, z, w}, Integers, 10] (*Calculate number of paths for each solution*) paths = 4!/(x! y! z! w!) /. sol (*Total number of valid paths*) total = Total[paths] (*Check if option A is correct*) Print["Is option A correct? ", total == 12] The above is the computational method I used to verify whether Option A is correct. Now I want to visualize all valid paths for each option using grid diagrams, count the number of paths and calculate the probabilities to determine the correctness of each option. How can I draw such diagrams? [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=2026-04-11_092228.png&userId=3593842 Bill Blair 2026-04-11T01:43:23Z https://community.wolfram.com/groups/-/m/t/3686348 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/e1aa2b28-96b1-4104-8347-1e38ecdf5e0d Brian Beckman 2026-04-14T16:21:26Z https://community.wolfram.com/groups/-/m/t/3688913 An Offset surface, is a smooth surface (without edges, peaks, or singular points, where each point has a tangent plane. Examples: Spheres, ellipsoids, toroids, cylinders and planes), it is defined as follows: sd[u, v] = s[u, v] + d n[u, v] and sd[u, v] = s[u, v] - d n[u, v] s[u, v] it is a smooth surface, d = distance, n[u, v] it is the normal unitary vector to the surface s[u, v]. In this notebook we will obtain the offset surface of the plane: {-2, 4, 5} + u{4, -4, -6} + v{5, 3, -5} with d = 3. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/fb6f7a2c-9212-43d4-902c-fd22f32c8bbe Alejandro Latorre Chirot 2026-04-15T19:42:55Z https://community.wolfram.com/groups/-/m/t/3688732 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/5a7894fc-4cd0-439f-9d83-25d28bb47b37 Ed Pegg 2026-04-15T15:43:34Z https://community.wolfram.com/groups/-/m/t/3688820 An Offset surface, is a smooth surface (without edges, peaks, or singular points, where each point has a tangent plane. Examples: Spheres, ellipsoids, toroids, cylinders and planes), it is defined as follows: sd[u, v] = s[u,v] + d n[u, v] and sd[u, v] = s[u,v] - d n[u, v] s[u, v] it is a smooth surface, d = distance, n[u, v] it is the normal unitary vector to the surfaces[u, v]. In this notebook we will obtain the offset surface of the cylinder: {Cos[u],Sin[u],v} with d = 1.5. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/88b265ae-7b15-4b35-aedd-0301a3453cc5 Alejandro Latorre Chirot 2026-04-15T15:29:44Z https://community.wolfram.com/groups/-/m/t/3688629 It is the circle that has the center to the incenter. To obtain ii first we will calculate the tangent points of the inscribed circumference and we will trace the Gergonne triangle which is what passes through those points. Then we will get the Gergonne point of the base triangle. By the Gergonne point we draw a parallel line to one side of the Gergonne triangle and we will obtain one of the points through which the Adams circumference passes. We will calculate the radius at that point and the center. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/ea497de1-cadd-4a10-9590-23b9e8a7ae73 Alejandro Latorre Chirot 2026-04-15T14:52:46Z https://community.wolfram.com/groups/-/m/t/3686271 An Offset surface, is a smooth surface (without edges, peaks, or singular points, where each point has a tangent plane. Examples: Spheres, ellipsoids, toroids, cylinders and planes), it is defined as follows: sd[u, v] =s[u, v] + d n[u, v] and sd[u, v] =s[u, v] - d n[u, v] s[u, v] it is a smooth surface, d = distance, n[u, v] it is the normal unitary vector to the surface s[u, v]. In this notebook we will obtain the offset surface of the toroid: {(5 + 2 Cos[u])Cos[v], (5 + 2 Cos[u])Sin[v], 2 Sin[u]} with d = 1.5. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/3930c95f-9d47-4065-8002-d47549c73e75 Alejandro Latorre Chirot 2026-04-14T17:45:18Z https://community.wolfram.com/groups/-/m/t/3686328 An Offset surface, is a smooth surface (without edges, peaks, or singular points, where each point has a tangent plane. Examples: Spheres, ellipsoids, toroids, cylinders and planes), it is defined as follows: sd[u, v] = s[u, v] + d n[u, v] and sd[u, v] = s[u, v] - d n[u, v] s[u, v] it is a smooth surface, d = distance, n[u, v] it is the normal unitary vector to the surface s[u, v]. In this notebook we will obtain the offset surface of the sphere: {3 Cos[u]Cos[v], 3 Sin[u]Cos[v], 3 Sin[v]} with d = 2. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/0ca7eedf-b306-449f-8ed2-39ee801127fe Alejandro Latorre Chirot 2026-04-14T14:01:10Z