Humanity in a 1000 years I: autoevolution

It is probably absurd trying to predict what will happen with humanity in a 1000 years. It is likely that even the smartest person living in 1000 C.E. would find it difficult to imagine the era we live in now.
Maybe armed with rationalism and the scientific method we are in a slightly better position to do it now. On the other hand, it is possible that the methods of thought or even the very apparatus of the mind will make so much progress that our descendands will create things we would never be able to dream of. In fact, I am (somewhat paradoxially) about to claim precisely that. At any rate, it appears almost inevitable the increase in body of human knowledge will lead to incredible changes in human society and the human way of life. Unless, of course, some terrible catastrope or a new dark age will prevent it.
If so, the task of imagining humanity in the year 3000 C.E. appears almost hopeless. Nevertheless, I still think it is worthwhile. Why? Because it is an amusing thought experiment. Because thinking about the future may change the future. Because trying to stretch our ability to predict or analyse to the limit may teach us something. Evef if it won't, it is still bound to be fun :-) Let me give a go at it, then!

I warn beforehand that my view of the future is somewhat optimistic. I am assuming humanity will not be destroyed by nuclear war, alien invasion, asteroid impact or any other calamity. I am also assuming scientific progress is not going to stop or reverse as a result of such an event. My entire "prediction" is something of a mixture between what I believe will happen and what I hope will happen.

Autoevolution

Charles Darwin taught us that humans are not essentially different from any other animal. Like any other animal, or indeed, any other living creature, we gradually evolved from other species over vast periods of time. The governing principle of that process is natural selection. The princinple is so obvious it is almost a tautology: the speciments most adapted to survival are more likely to survive so each generation is adapted better than the previous one. Add mutations into the mix and we get evolution.

How long does evolution continue? As far as we know it is indefinite. External conditions change, different species compete and, most importantly from my point of view, nature never reaches perfection. There are always improvements to be made.

Improvements? Isn't home sapiens sapiens perfect? Isn't it the peak of creation?

What on Earth gave us the arrogance to think that? Oh, sorry, I know what it is: evolution ;-)

Homo sapiens sapiens can and should be improved. There's no reason to think we can't be more healthy, more enduring, more intelligent. The problem is, we are different from other animals after all. We change our environment, adapting it to our needs. This process is much faster than the self-adaptation resulting from biological evolution. The result: natural selection, the driving force of biological evolution is no longer valid for this species.

Not only that we are (apparently) no longer becoming better, we are probably becoming worse. Random mutation introduce noise into our genetic code. In the same time, in a modern society (I mean the developed countries) "weak" individuals are not allowed to perished (which is a good thing!) and have no problem of spreading their genes. Anyone short of a Nazi would agree that the situation in which each individual of society is protected and able to satisfy her basic needs is a healthy one, from a moral stand point. The downside is that in the long run the human race faces physical and intellectual degeneration (I recommend the amusing comedy "Idiocracy" on this subject precisely).

Luckily, this threat, created, in a sense, by modern technology, finds its solution in the same source. In recent decades, the field of genetics experienced vast progress. The extent of the progress is such that genetic engineering has become possible. Now, we are only making our first step in this direction. However, we are discussing a problem that will only become relevant in a the very long run and there is little doubt that by that time our ability to manipulate the genetic codes of living being including ourselved will be perfected.

Thus, genetic engineering of human beings appear to me inevitable in order to avoid degeneration. However, we can and should go beyond this and apply genetic engineering in order to improve ourselves rather than merely preserving ourselves on the same level.

OK, I thought it is going to be one post, but it would take me ages to complete in this rate. So I'm posting the beginning, to be (hopefully) continued...

Regular Polytopes and Tilings

A few random thoughts on regular polytopes and tilings I wanted to share.

2D

Given m,n >= 3, we can try to build a 2-dimensional tiling where m regular n-gons meet at each vertex. The angle of a regular n-gon is alpha = (1 - 2/n) pi, and we have the following three cases:

1. m alpha < 2 pi. This yields a regular polyhedron (a Platonic solid). There are 5 cases like that:
   a. n = 3, m = 3: tetrahedron, a self-dual polyhedron, the 3-dimensional simplex
   b. n = 3, m = 4: octahedron, the 3-dimensional cross-polytope
   c. n = 3, m = 5: icosahedron
   d. n = 4, m = 4: cube, dual to octahedron, the 3-dimensional hypercube
   e. n = 5, m = 3: dodecahderon, dual to icosahedron
   Each of those defines a finite subgroup of SO(3), the 3-dimensional rotation group and of O(3) the 3-dimensional rotation-and-reflection group. These subgroups are, of course, the symmetry groups of the polyhedra.
2. m alpha = 2 pi. This yields a regular tiling of the Euclidean plane. There are 3 cases like that:
   a. n = 3, m = 6
   b. n = 4, m = 4 A self-dual tiling
   c. n = 6, m = 3 dual to a
   Each of those defines a discrete subgroup of the group of isometries (rotations, translations and reflections) of the Euclidean plane. Alternatively we can use orientation-preserving isometries (rotations and translations only).
3. m alpha > 2 pi. This yields a regular tiling of the hyperbolic plane. There's an infinite number of cases. Each of them defines a discrete subgroup of SO(2, 1). The later group has various geometric realizations:
   a. Orientation-preserving isometries of the hyperbolic plane.
   b. Lorentz transformations of special relativity in 3-dimensional spacetime (2 space dimensions and 1 time dimension).

3D

Given A, B regular polyhedrons, we can try to build a 3-dimensional tiling where #{faces of B} A-polyhedra meet at a B-type vertex. What do I mean by a B-type vertex? Imagine the vertex being in the ceter O of a B-polyhedron Y. Fix a face F of
Y. F corresponds to an A-polyhedron X of the tiling. The lines passing through the O and the vertices of F correspond to edges of X.
This wouldn't work for any A, B. For purely combinatorial reasons, we need

#{faces meeting at a vertex of A} = #{sides of a face of B}

Geometrically, we again have three cases, depending on

• alpha, the dihedral angle of A, that is, it is the angle between its two adjacent faces.
• m, the number of faces meeting at a vertex of B.

The three cases are:

1. m alpha < href="http://en.wikipedia.org/wiki/Convex_regular_4-polytope">polychoron). There are 6 cases like that:
   a. A = tetrahedron, B = tetrahedron: pentachoron, a self-dual polychoron. It is the 4-dimensional simplex.
   b. A = tetrahedron, B = octahedron: hexadecachoron. It is the 4-dimensional cross-polytope.
   c. A = tetrahedron, B = icosahedron: hexacosichoron.
   d. A = cube, B = tetrahedron: tesseract, dual to the hexadecahoron. It is the 4-dimensional hypercube.
   e. A = octahedron, B = cube: icositetrachoron, a self-dual polychoron.
   f. A = dodecahedron, B = tetrahedron: hecatonicosachoron, dual to the hexaicosohoron.
   Each of those defines a finite subgroup of SO(4), the group of 4-dimensional rotations. It also defines a finite subgroup of O(4), the group of 4-dimensional rotations-and-reflections.
2. m alpha = 2 pi. This yields a regular tiling of the Euclidean space. There is only 1 case like that: A = cube, B = octahedron. It defines a discrete subgroup of the group of isometries (rotations, translations and reflections) of the Euclidean space, or of the group of orientation-preserving isometries (no reflections).
3. m alpha > 2 pi. This yields a regular tiling of the 3-dimensional hyperbolic space. There are 4 cases like that:
   a. A = cube, B = icosahedron
   b. A = dodecahedron, B = octahedron: dual to a
   c. A = dodecahedron, B = icosahedron: self-dual
   d. A = icosahedron, B = dodecahedron: self-dual
   Each of those defines a discrete subgroup of SO(3, 1). The later group has various geometric realizations:
   a. The group of orientation-preserving isometries of the 3-dimensional hyperbolic space.
   b. The group of Lorentz transformations in special relativity.
   c. The group of orientation-preserving conformal transformations of the
   2-sphere.
   Realization b is intriguing since it makes me wonder whether these discrete subgroups appear in any physically-interesting situation.
   Realization c is intriguing for the following reason. Each such transformation has one or two fixed points. Consider the set of fixed points of all transformations belonging to a given discrete subgroup. This is a countable subset of the sphere, invariant under the discrete subgroup (due to conjugation). Clearly it must be either dense everywhere or a sort of fractal, but I don't know which.

4D

This time we take A, B to be regular polychorons. We want to construct a 4-dimensional tiling of A-polychorons which meet at a B-type vertex. The combinatorial compatibility condition is

vertex polyhedron of A = hyperface polyhedron of B

We have 3 geometric cases:

1. A regular tiling of the 4-sphere, that is, a 5-dimensional regular polytope. There are 3 cases like that:
   a. A = pentachoron, B = pentachoron: the self-dual 5-dimensional simplex.
   b. A = pentachoron, B = hexadecahoron: the 5-dimensional cross-polytope.
   c. A = tesseract, B = pentachoron: the 5-dimensional hypercube, dual to a.
2. A regular tiling of the 4-dimensional Euclidean space. One example is
   A = tessaract, B = hexadecahoron, which is self-dual
3. A regular tiling of the 4-dimensional hyperbolic space.

There are 7 exotic objects (that is, object special to dimension 4) among cases 2-3:

1. A = pentachoron, B = hexacosichoron
2. A = hexadecachoron, B = icositetrachoron
3. A = tesseract, B = hexacosichoron
4. A = icositetrachoron, B = tesseract: dual to 2
5. A = hecatonicosachoron, B = pentachoron: dual to 1
6. A = hecatonicosachoron, B = hexadecachoron: dual to 3
7. A = hecatonicosachoron, B = hexacosichoron: self-dual

At the moment I'm not sure which of them is a tiling of 4-dimensional Euclidean space and which is a tiling of 4-dimensional hyperbolic space.

Higher dimension

We take A, B to be regular n-dimensional polytopes. We want to construct an n-dimensional tiling of A-polytopes which meet at a B-type vertex. The combinatorial compatibility condition is

n-1-dimensional vertex polytope of A = n-1-dimensional hyperface polytope of B

Once again, we have 3 geometric cases:

1. A regular tiling of the n-sphere, that is an n+1-dimensional regular polytope. There are 3 cases like that:
   a. A = n-dimensional simplex, B = n-dimensional simplex. This is the self-dual
   n+1-dimensional simplex.
   b. A = n-dimensional hypercube, B = n-dimensional simplex. This is the n+1-dimensional hypercube.
   c. A = n-dimensional simplex, B = n-dimensional cross-polytope. This is the
   n+1-dimensional cross-polytope, dual to the n+1-dimensional hypercube.
2. A regular tiling of the n-dimensional Euclidean space. There is only 1 case:
   A = n-dimensional hypercube, B = n-dimensional cross-polytope.
3. A regular tiling of the n-dimensional hyperbolic space. There are none!

Indefinite signature

As we said, we construct n-dimensional tilings out of a pair A, B of n-dimensional polytopes. Now, a polytope is a tiling of the n-1-sphere. What if we take A, B to be tilings of the n-1-dimensional hyperbolic space instead? Logically, we should get a tiling of a space of Lorentzian signature, since the hyperbolic space plays the same role in Minkowski space that the sphere plays in Euclidean space. I'm not sure how would such a tiling look like, it appears it would be
self-interecting. As before, such tilings would come with different curvatures. That is, we should get

1. Positive curvature: a tiling of de Sitter space.
2. Zero curvature: a tiling of Minkowski space.
3. Negative curvature: a tiling of anti-de Sitter space.

If it indeed makes sense, we would also get discrete subgroups of the symmetry groups of the aforementioned spaces. For instance, we might get a discrete subgroup of the symmetry group of Minkowski space: the Poincare group. I wonder whether there exists a physical object with this kind of symmetry group: a sort of relativistic crystal!