There is one
outstanding genius of the past who is still almost unknown. His name is Ludwig
Schlaefli and his achievement, unrecognised during his lifetime, defies belief.
He worked out the number of possible regular shapes which could exist in four
dimensions. That may not sound much, particularly if you start by considering
the equivalent problem in, say, two dimensions.
On a flat,
two-dimensional piece of paper, it is obvious that you can draw regular shapes
with three, four, five, any number of sides you like, triangles, squares,
pentagons, hexagons and so on, right up to infinity-gons. You might think the
same would be true in three dimensions. It isn't. As Lewis Carroll pointed out,
in 3D the regular shapes are "provokingly few in number", only five
of them. Plato and the ancient Greeks knew this and we call them the Platonic
solids (tetrahedron, cube, octahedron, dodecahedron and icosahedron with respectively
4, 6, 8, 12 and 20 faces). You can't create any more, however hard you try.
It's fairly easy to
think in three dimensions and to convince yourself that the five Platonic
solids are the only ones possible. It's much harder to think in four
dimensions. In the 1800s, many very distinguished mathematicians tried to do
just that and to show which regular 4D shapes existed and what they looked
like. Schläfli, with a truly astonishing and genius-level feat of mental
gymnastics, beat the lot of them to it. Yet he died with this amazing
achievement completely unrecognised.
See the comment below for a brief biography.
Ludwig Schläfli (1814–1895) was born in Bern, Switzerland.
ReplyDeleteAlthough he was only 15 when he entered Gymnasium, Schläfli was already studying the differential calculus. At the age of 17 he began to study theology at the Academy in Bern. After graduating in 1836 he became a schoolteacher of mathematics and science at the Burgerschule in Thun. He worked there for 10 years, studying advanced mathematics in his spare time and attending Bern University one day a week. He was a natural linguist and spoke many languages including French and Italian.
In Bern, Schläfli met the geometrician Jakob Steiner who was impressed with his language skills and his mathematical knowledge. Steiner was to visit Rome with two mathematicians from Berlin. He recommended Schläfli as a travel companion, saying that he was a provincial mathematician and not very practical but that he learned languages like child's play. So Schläfli accompanied them to Rome as interpreter. He learnt a lot from discussions with these leading mathematicians, in particular having a daily lesson in number theory from Lejeune Dirichlet.
After six months in Italy Schläfli returned to his job in Thun, continuing to correspond with Steiner. He took a teaching post at the University of Bern in 1848, surviving on a very low salary (“I was confined to a stipend of Fr 400 and, in the literal sense of the word, had to do without”) until 1853 when he was promoted to extraordinary professor. He worked on two major mathematical topics – elimination theory and the “Theory of multiple continuity”, for which his key work of 1852 begins:
“This treatise, which I have the honour of presenting to the Imperial Academy of Science, is an attempt to found and to develop a new branch of analysis that would, as it were, be a geometry of n dimensions, containing the geometry of the plane and space as special cases for n = 2, 3. I call this the "theory of multiple continuity" in the same sense in which one can call the geometry of space that of three-fold continuity. As in that theory, a 'group' of values of coordinates determines a point, so in this one a 'group' of given values of the n variables will determine a solution.”
The treatise was a long one and it was rejected by both the Austrian Academy of Sciences and the Berlin Academy of Science. The complete work was only published in 1901, after his death, and only then did its importance become fully appreciated.
In it, Schläfli defines polytopes (he uses the word polyschemes) as higher-dimensional analogues of polygons and polyhedra. He introduced what is today called the Schläfli symbol. It is defined inductively. {n} is a regular n-gon, so {4} is a square. Then {4, 3} is the cube, since it is a regular polyhedron with 3 squares {4} meeting at each vertex. Then the 4-dimensional hypercube is denoted as {4, 3, 3}, having three cubes {4, 3} meeting at each vertex. Euclid, in the Elements, proves that there are exactly five regular solids in three dimensions. Schläfli proves that there are exactly six regular solids in four dimensions {3, 3, 3}, {4, 3, 3}, {3, 3, 4}, {3, 4, 3}, {5, 3, 3}, and {3, 3, 5}, but only three in dimension n where n ≥ 5, namely {3, 3, ..., 3}, {4, 3, 3, ....,3}, and {3, 3, ...,3, 4}.
Schläfli made an important contribution to non-Euclidean (elliptic) geometry when he proposed that spherical three-dimensional space could be regarded as the surface of a hypersphere in Euclidean four-dimensional space. Other investigations covered topics such as partial differential equations, the motion of a pendulum, the general quintic equation, elliptic modular functions, orthogonal systems of surfaces, Riemannian geometry, the general cubic surface, multiply periodic functions, and the conformal mapping of a polygon on a half-plane.
During his lifetime Schläfli never received full credit for his remarkable achievements.
(With acknowledgement to the article by J J O'Connor and E F Robertson in the MacTutor History of Mathematics archive.)
Rotation in four-dimensional space.
ReplyDeletehttps://youtu.be/vN9T8CHrGo8
The 5-cell is an analog of the tetrahedron.
https://youtu.be/z_KnvGGwpAo
Tesseract is a four-dimensional hypercube - an analog of a cube.
https://youtu.be/HsecXtfd_xs
The 16-cell is an analog of the octahedron.
https://youtu.be/1-oj34hmO1Q
The 24-cell is one of the regular polytope.
https://youtu.be/w3-TqPXKlVk
The hypersphere is an analog of the sphere.
WE SEEM TO LIVE IN A WORLD OF 3 DIMENSIONS
ReplyDeleteThis blog tells us about the 4th spatial dimension. Indeed I think it assumes there are infinitely many spatial dimensions. So what is special about our world that we seem to have just 3?
Or is the 3-ness of our world an illusion?
The blog refers to a creature (an intelligent pond skater, say) that lives on a flat plane (the surface of its pond). Presumably, as the blog puts it, our imaginary creature “would understand length and width, but height would be unknown – a mysterious closed book”.
Now let’s suppose our pond skater, in the course of its evolutionary history, had acquired a degree of intelligence – albeit an intelligence entirely related to its history of living on the surface of water. To it, the world seems to have just 2 dimensions. Could we help it to understand that there are “really” 3 dimensions? Or is its evolutionary history such that its mind is stuck with the perception of only 2?
Human minds have evolved in a world where height was important, as well as length and breadth. But as far as we can imagine, the 4th dimension was not important in our evolution. Does that very history limit our perception of the 4th dimension?
In other words, is the 3-ness of our perception an illusion, in the same sense as the pond skater’s inability to perceive the 3rd dimension can be considered an illusion?
Incidentally, what would we mean if we thought the 4th dimension had been “important” in our evolutionary history?