Equidistant Points on a Sphere: What is the Series and How to Draw Efficiently?

  • Context: Graduate 
  • Thread starter Thread starter ChemistryInclined
  • Start date Start date
  • Tags Tags
    Geometry Sphere
Click For Summary

Discussion Overview

The discussion revolves around the challenge of placing equidistant points on a sphere, specifically seeking a series of configurations that achieve this for a number of points greater than 40. Participants explore mathematical concepts related to point distributions on a sphere, including references to Platonic solids and the geometric properties that govern these arrangements.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks to understand the series of equidistant points on a sphere, starting from known configurations like 2, 4, and 8 points, and asks for the next configurations leading up to 40 points.
  • Another participant challenges the claim that points on a cube (8 points) are equidistant, noting that diagonal points are further apart than those on the same face.
  • A different participant clarifies that the requirement is for nearest points to be equidistant, suggesting that the criteria can only be met with configurations up to a pyramid shape.
  • Participants express interest in the concept of Platonic solids, noting that existing descriptions only cover up to the icosahedron, which may not suffice for the desired number of points.
  • There is a concern about how to physically represent these points on a sphere, particularly when considering the practicalities of marking points on the inside of a hollow sphere.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the configurations of equidistant points, with multiple competing views on the feasibility and definitions of equidistance among points on various geometric shapes.

Contextual Notes

Participants mention limitations in existing resources regarding the configurations of equidistant points beyond the icosahedron, indicating a potential gap in available mathematical frameworks for higher numbers of points.

Who May Find This Useful

This discussion may be of interest to mathematicians, physicists, and engineers exploring geometric configurations, as well as hobbyists interested in practical applications of geometry in physical models.

ChemistryInclined
Messages
2
Reaction score
0
Hello all,

apologies if this is obvious, but searching the internet did not provide me with a satisfactory answer.

I am trying to design a sphere with 40+ points on it, which are all equidistant.
To rephrase, what is the number series of points on spheres that are equidistant to each other?
To begin with, the series is 2 (one point at the top, one at the bottom), then 4 (equidistant pyramid), 8 (cube), then ... what follows?
What is the closest number to 40 (or whatever) where all are equidistant, and what are the angles? Illustrations welcome. No buckminster fullerenes please or whatever (since they don't fulfill this criterium), what I desire here is that all points on the sphere have an equal distance to the next.

Any budding or professional mathematicians who could help me? I imagine Riemann or the likes have worked this out... it'd be awesome to see the mathematics behind the deduction of this problem.

Regardless, I am after the real thing. How can one draw this most efficiently on a physical sphere? Preferably on the inside of a hollow sphere (otherwise I'll have to stick needles through the points drawn on the outside of the sphere)?

Any input is appreciated. Thanks.
 
Physics news on Phys.org
I can't really answer your question, but forming a cube with 8 points does not make them all equidistant from one another the ones on diagnols from one another are further apart than the ones that arent on a diagnol line from them if that makes any sense.
 
No, the issue is that all *nearest* points are equidistant to each other. If it had to be equidistant to ALL points, the criterium could be only fullfilled up to the pyramid!

Wow, I stumbled upon a new word: Platonic Solids!

Sadly the descriptions I find are only up to the icosahedron... so by far not enough for my plans.

And I am still dumbfounded as to how to draw points of even a pyramid on a real physical sphere!
 
ChemistryInclined said:
Wow, I stumbled upon a new word: Platonic Solids!

Sadly the descriptions I find are only up to the icosahedron... so by far not enough for my plans.

There are only five Platonic solids: the tetrahedron (triangle-base pyramid), the cube, the octohedron, the dedecahedron, and the icosahedron.
 

Similar threads

Undergrad Equidistance of Points in a Sphere?
  • · Replies 5 ·
Replies
5
Views
4K
Graduate What is the relationship between Pi and curvature on different surfaces?
  • · Replies 21 ·
Replies
21
Views
9K
Graduate What are Some Unresolved Questions in Electrostatics?
  • · Replies 9 ·
Replies
9
Views
6K
  • Poll Poll
Geometry A Comprehensive Introduction to Differential Geometry series by Spivak
  • · Replies 15 ·
Replies
15
Views
23K
Graduate What Does Mathematical Sameness Mean?
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Attempt to visualize higher dimensions
  • · Replies 25 ·
Replies
25
Views
6K
Graduate Ball Lightning Debunk, New Proposal
  • · Replies 1 ·
Replies
1
Views
4K
Mathematica This Week's Finds in Mathematical Physics (Week 264)
  • · Replies 6 ·
Replies
6
Views
5K
Programs How good is this applied math program?
  • · Replies 13 ·
Replies
13
Views
4K
  • Sticky
Kerbal Space Program Tutorial Series - Open Access to 1.7
  • · Replies 48 ·
2
Replies
48
Views
69K