area ← dim ##.ksphere radius ⍝ Hypersphere surface area.
Inspired by Eugene McDonnell's paper [1], [ksphere] returns the surface area of
an ⍺-sphere of radius ⍵.
NB: Wolfram [4] points out: "Unfortunately, geometers and topologists adopt in-
compatible conventions for the meaning of "n-sphere," with geometers referring
to the number of coordinates in the underlying space ("thus a two-dimensional
sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the
dimension of the surface itself". [ksphere] sides with the topologists so, here,
we call a circle a 1-sphere or S1.
Here are the surface areas of some unit hyperspheres. In particular, the "sur-
face area" of a 1-sphere (circle) is its circumference.
(1 to 10) ksphere 1 ⍝ surface of unit ⍺-spheres.
6.2832 12.566 19.739 26.319 31.006 33.073 32.47 29.687 25.502 20.725
Notice that the area of a unit k-sphere achieves a maximum at around 6 dimens-
ions. In his paper, Eugene poses the question: What dimension of space gives the
maximum hypervolume to the unit radius hypersphere?
As coded, [ksphere] is a continuous function over real ⍺, so we can investigate
non-integral dimensions. Here is a crude Newton-Raphson technique for finding a
local maximum.
max←{⍺←⎕CT*÷2 ⍝ Local maximum using Newton-Raphson.
∆x←1+¯1 0 1×⍺ ⍝ x deltas.
⍺⍺{ ⍝
∆∆x←⍵×∆x ⍝ x-∊ x x+∊
∆∆y←⍺⍺ ∆∆x ⍝ f(x-∊) f(x) f(x+∊)
d1←÷⌿¯2-/↑∆∆y ∆∆x ⍝ first difference.
d2←÷/-/↑d1(2↑∆∆x) ⍝ second difference.
∆←⌈/d1÷d2 ⍝ increment f'(⍵)÷f"(⍵)
∆=0:⍵ ⍝ approx convergence: done.
∇ ⍵-∆ ⍝ ⍵ → ⍵ - f'(⍵)÷f"(⍵)
}⍵
}
Then:
ksphere∘1 max 6 ⍝ approx maximum surface area of unit k-sphere.
6.256946087
Seaching the Internet shows the maximum to be at dimension 6.2569464048605768 to
17 sig figs. Wolfram Alpha [5] is particularly good for this sort of exercise.
Notice that the volume of the (k+1)-ball inside the k-sphere is easily derived
from its surface area:
kvol ← {⍵×((⍺-1) ksphere ⍵)÷⍺} ⍝ volume of ⍺-ball of radius ⍵.
So to answer Eugene's question:
kvol∘1 max 5 ⍝ approx maximum volume of unit k-sphere.
5.256946138
(muse:
Visualising Hyperspheres
------------------------
In general, we can construct n-sphere Sn by gluing together the surface
(n-1)-spheres of two n-balls. Let's start with a familiar 2-sphere S2:
Take a pair of 2-balls (flat circular discs) of thin rubber sheet and care-
fully glue their outside (S1) edges together. Then inflate the enclosed
space to produce a regular 2-sphere. A Flatlander [7], living on the surface
of S2, would perceive it as an unbounded, though finite, 2-universe.
Ferdinand Magellan was in a similar situation as he explored his finite but
unbounded 2-sphere.
Similarly, S1 may be constructed by gluing the endpoints of a pair of 1-
balls (line segments) together and bowing the lines outwards to form a
circle. A "Linelander", travelling around the circle, would not notice the
two joining S0 points.
We "3-landers" could build a 3-sphere by gluing the 2-sphere boundaries of a
pair of "adjacent" 3-balls (though we'd need to borrow a little 4-space in
which to do the job). To make it easier, let's don scuba gear and swim in-
side one of a pair of massive 3-balls (regular 3D balls) of sea-water, which
are suspended close to each other in 4-space. After our 4-lander friend has
glued together the outer 2-sphere-boundaries of our 3-balls (using special
transparent sea-water adhesive) we can swim from our home hemiball, straight
ahead in any direction, into the other hemiball. If the sea-water glue
really is transparent, we should not notice as we swim through the 2-sphere
join. Again, we're in an unbounded manifold: even though we are within a
finite volume of water, no matter how far we swim in any direction, we will
never encounter a boundary.
NB: Before using a harpoon gun, please heed the warning, at the end of the
notes on →life←, about using artillery in a finite manifold.
The Poincaré Conjecture
-----------------------
Imagine a 2-dimensional (Flatland) spider wandering around in the surface of
a large soap bubble. To amuse herself in this bleak landscape she plays a
little game: as she moves, she extrudes a single filament of web, which is,
of course, also embedded in the surface of the S2 bubble. Her game is to
roam around her 2-sphere looking for the starting end of the filament and,
when she finds it, to reel it in. As she is holding both ends of the fil-
ament, topologically speaking, it forms a circle S1. Our spider finds that
she can _always_ reel in her web-loop and so she convinces herself that she
must be in the surface of a sphere, rather than, for example, in the surface
of a more exotic 2-manifold, such as a torus (S1×S1).
We can play the same game in our sea-water 3-sphere: we swim forwards while
uncoiling our wreck-diving rope, leaving one end at a fixed position in the
3-water. We find that, after swimming straight ahead in any direction for a
distance of 2×D, where D is the diameter of our 3-spheres, the end of the
rope hoves into view again. Now, if we grab both ends of the rope and start
to pull, we should be able to reel in the whole of the loop without its be-
coming tight.
By analogy with the flat-spider in the surface of torus S1×S1, we might not
always be able to reel in our rope had we found ourselves swimming, for ex-
ample inside S1×S1×S1, which is made by gluing together opposite faces of a
_cube_ of sea-water.
It has long been assumed that a 3-sphere is the only 3-manifold in which one
can always reel in the rope-loop in this way. This assumption is known as
the "Poincaré Conjecture" and a proof of it, which carried a million-dollar
reward, eluded mathematicians for the whole of the twentieth century. The
conjecture was proved in 2002 by Grigori Perelman, building on work by Rich-
ard Hamilton. See ref[8] below.
Explore 3-manifolds by downloading this magnificent hyper-flight-simulator:
http://www.geometrygames.org/CurvedSpaces/index.html
(muse: At a picnic, never let a topologist slice the loaf.)
)
Technical note:
As [ksphere] utilises only scalar pervasive functions, it is itself scalar per-
vasive. This means that it may be applied directly between conformable arguments
of higher rank and depth.
Refs:
[1] http://www.jsoftware.com/papers/eem/storyofo.htm
[2] http://en.wikipedia.org/wiki/Deriving_the_volume_of_an_n-ball
[3] http://en.wikipedia.org/wiki/Hypersphere#cite_note-0
[4] http://mathworld.wolfram.com/Hypersphere.html
[5] http://www.wolframalpha.com
[6] http://www.geometrygames.org/CurvedSpaces/index.html
[7] http://en.wikipedia.org/wiki/Flatland
[8] http://en.wikipedia.org/wiki/Poincare_conjecture
Examples:
⎕pp←5
⍝ 1-sphere is a circle:
1 ksphere 10 ⍝ circumference of circle, radius 10.
62.832
2×○ 10 ⍝ compare with: 2×Pi×r.
62.832
⍝ 2-sphere is a reqular sphere in 3-space:
2 ksphere 10 ⍝ surface area of sphere, radius 10.
1256.6
4×○ 10*2 ⍝ compare with: 4×Pi×r-squared.
1256.6
(0 to 10) ksphere 1 ⍝ surface of unit ⍺-spheres.
2 6.2832 12.566 19.739 26.319 31.006 33.073 32.47 29.687 25.502 20.725
kvol ← {⍵×((⍺-1) ksphere ⍵)÷⍺} ⍝ volume of ⍺-ball of radius ⍵.
(0 to 10) kvol 1 ⍝ volumes of unit ⍺-balls.
2 3.1416 4.1888 4.9348 5.2638 5.1677 4.7248 4.0587 3.2985 2.5502 1.8841
See also: kball to life
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