Spheres| Spheres |
Topological > s.a. complex
structures; differentiable
manifolds [exotic
Sn].
* Annulus conjecture:
If S and S' are
two disjoint locally flat (n–1)-spheres in Sn,
the closure of the region between them is homeomorphic to Sn–1 ×
[0,1].
* Smith conjecture (theorem):
The background is that the set of fixed points of a periodic homeomorphism
S3 → S3 is 
S1;
Then this circle cannot be knotted; Has been shown to be true if the homeomorphism
is
smooth
enough.
@ References: Peterson AJP(79)
[visualizing]; Morgan & Bass ed-84 [Smith
conjecture]; Chen & Lin JDG(01)
[scalar curvature]; Schueth JDG(01)
[S3, isospectral].
> Topics: see Brouwer
Theorem; euler number; Hopf
Fibration; Smale Conjecture; types
of manifolds [parallelizable].
Metric > s.a. integration; laplacian; trigonometry [spherical].
* S1: A possible parametrization is x = 
(t2–1)/(t2+1), y =
2t/(t2+1);
covers half the circle for t 
[–
,].
* S2:
If a is the radius of the sphere (as embedded in flat space), the
circumference of a circle of radius r on the sphere, and the surface
area of the spherical cap enclosed by it are, respectively,
C(r) = 2
a sin(r/a) , A(r)
= 2r2 [1–cos(r/a)]
.
(For an approximation,
cut 1 triangle out of hexagons and paste together to get
icosahedron; Add 1 triangle to get a pseudosphere).
* Complex dyad on S2:
There can be no non-vanishing vector field on S2,
let alone an orthonormal dyad in the ordinary sense, but a complex dyad (ma,
m*a) satisfying ma · ma =
0, m*a · m*a =
0, ma · m*a =
1, can be defined by (
a and 
a are
unit vectors)
ma = 2–1/2 exp{i cos}(a +
i a) , m*a =
2–1/2 exp{–i cos}(a –
i a)
.
* S3: In a 3-sphere of radius of curvature a, the volume of a ball of radius r is
V(B3) = 4a3 {
arcsin(r/a)
– (r/2a) [1 – (r/a)2]1/2}
(4r3/3)
[1 + O(r/a)2] .
* Sn: Area and scalar curvature of (n–1)-surface, and volume of interior n-ball:
S(Sn) =
2(n+1)/2 / 
[(n+1)/2]
; R(Sn)
= n(n–1) ;
V(Bn) = S(Sn–1)/n =
2n/2/[n(n/2)]
= (2)n/2/n!!
for n even,
2(2)(n–1)/2/n!!
for n odd.
@ References: Dowker CQG(90) [volume-preserving diffeomorphisms on S3]; Abdel-Khalek mp/00 [S7]; Boya et al RPMP(03)mp/02 [volumes].
Shere Packings
* Kepler's conjecture:
In R3, the usual packing
(which fills about 74% of the total available space) is the tightest
one;
The proof given in 1998 by Thomas Hales (after a proof "outline" published
in 1993 by W-Y Hsiang) relied on computer use; In 2004 AM after an
exhausting
reviewing process (editor MacPherson likened it to proofreading a
phone
book) decided to publish the analytical parts only, while the rest will
be
published in DCG.
* Random or amorphous packing:
Packing fractions are only
about 64%
of the total available space (this is for spheres, while spheroids – like
M&Ms
– can randomly pack more densely to fill between 68 to 71% of the total
available
space, and cigar-shaped ellipsoids could be randomly packed
with a density of almost 74%).
* Apollonian or osculatory
arrangement: An arrangement of d-dimensional spheres, each one
of
which touches d + 1 others.
@ General references: Moraal JPA(94);
Weaire 99 [I]; Aste & Weaire 00 [I]; Hales AM(05)
[proof of Kepler's conjecture + resources].
@ Random packings: Shlosman & Tsfasman mp/00; Radin JSP(08)
[phase transition].
Related Concepts and Spaces > s.a. lie
groups; Quaternions; spherical
harmonics; spherical
symmetry; Trinion.
* Spherical space: A manifold of the type Sn/H, H a
finite group acting
freely on Sn.
@ Monge metric: Zyczkowski & Slomczynski JPA(01)qp/00 [and
quantum states]; > s.a.
types of distances.
@ Geometry: J C Polking's spherical site;
> s.a. trigonometry.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
13 jun 2008