6 reputation
3
bio website slyman.org/blog/category/engineering-maths
location Huddersfield, United Kingdom
age
visits member for 2 years, 2 months
seen Jun 5 at 5:25

Relevant qualifications:

  • M.A. Computer Science (Cambridge University)
  • 2× A-Level Maths (includes pure, statistics, mechanics).

Mar
25
comment Conventional ordering of faces of regular polyhedron?
On closer examination: there are ways in which Professor Tegmark's 1996 paper offers superior insights: 1. His more complex "pixellation" is more area-uniform (his encoding includes an extra mathematical step for this purpose, that I might optionally adopt); 2. His "pixels" are more circular. ~~~ Whether one would choose his method or mine then comes down to a question of priorities: do you want to encode and present your data in a computationally efficient & geometrically simple manner; or do you require a very area-uniform solution that minimizes "pixels" almost regardless of encoding cost?
Mar
25
comment Conventional ordering of faces of regular polyhedron?
[Apologies that I don't have the "reputation" necessary to vote your answer as "helpful".]
Mar
25
comment Conventional ordering of faces of regular polyhedron?
...SO (to recap with the pertinent questions)... Can anyone formally prove #1 or #2 (from my previous comment)? Can we classify some numbering "curves" as more valid or more simple than others? (Note that my numbering scheme is not selected 100% arbitrarily, but gives a few nods to engineering and geometry as originally hinted...) Finally, will anyone mind me coining this convenient term "icosamap" as a descriptive label for a specific subset of the golf-ball mathematics that has been examined previously by better mathematicians?
Mar
25
comment Conventional ordering of faces of regular polyhedron?
The articles you cite are very interesting, but to my intuitive reading, only appear to support my original conclusions: 1. There is no possibility of improving upon the degree of 2D/3D distortion present in my "icosamap"; 2. There is no possibility (within the bounds of an icosahedral scheme, which is the best for reason #1) of a simpler or more uniform recursion than that which I am suggesting...
Mar
25
comment Conventional ordering of faces of regular polyhedron?
The variety in space-filling curves you listed informally confirms the consensus opinion here, that the choice of ordering is (in some degree) arbitrary (given at least, consistency and simplicity in recursion). However, on another thread, you quote Max Tegmark thus: "Unfortunately, it is a well-known group theoretical result that there are no completely regular point distributions on the sphere for N > 20." -- You're teasing me now! When I first asked this question, I entertained the vague notion that there might be some aspect of GROUP THEORY that would answer my question at least in part...
Mar
25
awarded  Commentator
Mar
25
comment Conventional ordering of faces of regular polyhedron?
I give it as my opinion (guess) that schemes based on triacontahedra or stellated/-akis polyhedra are only going to increase problems with uniformity, while failing to address the fundamental problem of distortion (after all, distortion in some degree is essential to any 2D/3D mapping.) However anyone wishing to satisfy themselves with some formal calculations to support my suggestion might start here: slyman.org/… -- This page contains a pyramid angle calculator and relevant general analytical solution.
Mar
25
comment Conventional ordering of faces of regular polyhedron?
...Naive improvements to these schemes might start with the "truncated icosahedron" (classic soccer ball shape, elegant for its 2^5=32 faces, concave semiregularity etc.); but we might demonstrate the similarity of distortion in each case by observing recursive subdivision at the basic vertices (as sub-vertex spacings approach zero separation). With some vertices of edge-order five, and some of order six; all such schemes are going to suffer from similar problems.
Mar
25
comment Conventional ordering of faces of regular polyhedron?
I would intuitively concur (with reference to the basic angular geometry of triangles, pentagons and hexagons) that the amount of 2D/3D projection distortion in Tegmark's scheme, the "geodesic grids" and my own "icosamap" scheme are all essentially equivalent...
Mar
25
comment Conventional ordering of faces of regular polyhedron?
This is the first time I've heard about Max Tegmark or his work (which resembles what I'm seeing elsewhere under the general heading, "Geodesic grid", after you signposted me to that). My comments on distortion applied strictly to the COBE sky cube (correctly mentioned in Tegmark's abstract as an inferior precursor to his/my work), or further to any similar scheme based on anything other than an icosahedral basis.
Mar
22
comment Conventional ordering of faces of regular polyhedron?
@DavidCary: Interesting articles, HOWEVER: the quadrilateralized spherical cube or "COBE sky cube" involves greater distortion either in area or geometry (note the curvilinear projection employed); and the "geodesic grid (Wikipedia)" is computationally and geometrically far more complex than what I am proposing (my "icosamap" involves only projected equilateral triangles, with simple and uniform recursion). I've considered other methods: look up "Rhombic triacontahedron" and consider subdividing each face into two non-equilateral triangles; but I still think my "icosamap" is best. Other ideas?
Aug
6
comment Conventional ordering of faces of regular polyhedron?
@GerryMyerson: Thank you for the hint! I didn't think I had the necessary account privileges — I'm not sure why I couldn't figure this out previously! I'm still relatively new around here.
Aug
6
revised Conventional ordering of faces of regular polyhedron?
Simplify title
Jul
28
comment Conventional ordering of faces of regular polyhedron?
@GerryMyerson: Do you think this post would get more attention if we simplified the title to: "Conventional ordering of faces of regular polyhedron?" Can this be done?
Jul
10
comment How to transform a set of 3D vectors into a 2D plane, from a view point of another 3D vector?
Please note that this is not a trivial topic. There are many surprising results to catch out newcomers; for example, when you project a 3D straight line onto a 2D screen from the perspective of a pinhole camera, the result is usually a 2D curve!
Jul
10
answered How to transform a set of 3D vectors into a 2D plane, from a view point of another 3D vector?
Jul
10
revised Conventional ordering of faces of regular polyhedron?
Updated URL for cited page as per automated request from administrator at progonos.com - and made a few minor grammatical and technical improvements.
Jun
22
comment Conventional ordering of faces of regular polyhedron?
22 days without an answer! @GerryMyerson - Is that a record for a geometry question that looks like it ought to be reasonably straightforward for someone with a degree in pure mathematics? Should I conclude that there simply aren't any conventions yet for ordering the faces of a polyhedron?
Jun
13
comment Conventional ordering of faces of regular polyhedron?
You're behind the times - we're now up to 23! Thank you for your attention to this thread, and for noticing my tendency toward perfectionism! As long as there weren't any comments or answers yet, I wanted to make sure people were working on the basis of the latest information, or on the basis of the best-informed designs I could muster without a better education in maths (my understanding has developed slightly over the last few weeks).
Jun
13
revised Conventional ordering of faces of regular polyhedron?
Punctuation, formatting, logical order, completeness