User:Dc.samizdat/Rotations

David Brooks Christie

dc@samizdat.org

June 2023 - November 2024

Abstract: The physical universe is properly visualized as a Euclidean space of four or more orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere with these objects distributed on its surface; the black hole at its center is the 4-ball of empty space it surrounds. The observable universe of galaxies may be visualized as a 4-sphere expanding radially from an origin point at velocity $c$, the propagation speed of light through 3-space, which is also the invariant velocity of all mass-carrying objects through 4-space. The propagation speed of light through 4-space $c_{4}$ is actually $c<c_{4}<2c$. This view is compatible with the theories of special and general relativity, and with the quantum mechanical atomic theory. It explains those theories as expressions of intrinsic symmetries.

Symmetries

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the mathematics of groups.

As I understand Noether's theorem (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than Einstein's relativity or Darwin's evolution or Euclid's geometry. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of the distinct symmetry groups. Thus all fundamental systems in physics, as examples quantum chromodynamics (QCD) the theory of the strong force binding the atomic nucleus and quantum electrodynamics (QED) the theory of the electromagnetic force, each have a corresponding symmetry group theory of which they are an expression. As I understand Coxeter group theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a Euclidean space of four dimensions, that is, they are four-dimensional Euclidean geometry. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the atom seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their group actions, including centrally rotations in 4-dimensional Euclidean space.

The geometry of the atomic nucleus

In Euclidean four dimensional space, an atomic nucleus is a 24-cell, the regular 4-polytope with 𝔽₄ symmetry. Nuclear shells are concentric 3-spheres occupied (fully or partially) by the orbits of this 24-point regular convex 4-polytope. An actual atomic nucleus is a rotating four dimensional object. It is not a rigid rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any nucleon number contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the kinematics of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly Euclidean in four dimensional space.

Rotations

The isoclinic rotations of the convex regular 4-polytopes are usually described as discrete rotations of a rigid object. For example, the rigid 24-cell can rotate in a hexagonal (6-vertex) central plane of rotation. A 4-dimensional isoclinic rotation (as distinct from a simple rotation like the ones that occur in 3-dimensional space) is a diagonal rotation in multiple Clifford parallel central planes of rotation at once. It is diagonal because it is a double rotation: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a twisted helical circle, rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, all the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. Clifford parallel planes are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the 3-sphere. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise detailed description enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: completely orthogonal planes, Clifford parallelism and Hopf fiber bundles, isoclinic geodesic paths, and chiral (mirror image) pairs of rotations, among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. The 6 regular convex 4-polytopes have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. Four dimensional Euclidean space is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of sensory experience in 4-dimensional space to draw upon.

For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: for example, all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles. But that is only the simplest case. Kinematic 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell.

To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. For example, if we imagine just 8 point-objects, evenly spaced around the 24-cell at the 8 vertices that lie on the 4 coordinate axes, and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time.

That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-orthoplex or 16-cell) performing the characteristic rotation of the 24-cell. But we can also imagine combining distinct rotations. What happens when multiple point-objects are orbiting at once, but do not all follow the Clifford parallel paths characteristic of the same distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of kinematic polytopes do they trace out, and how do their component parts relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of kinematics, and where dynamic stabilites arise, of kinetics.

Isospin

A nucleon is a proton or a neutron. The proton carries a positive net charge, and the neutron carries a zero net charge. The proton's mass is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet (I = 1/2). In isospin space, neutrons can be transformed into protons and conversely by actions of the SU(2) symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds.

According to the Noether theorem, isospin is conserved with respect to the strong interaction. Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space.

Isospin was introduced as a concept in 1932 by Werner Heisenberg, well before the 1960s development of the quark model, to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, Eugene Wigner introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated. Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections I₃ = +1/2 and −1/2 respectively.

Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a simple rotation in any space (of any number of dimensions). However, it does seem to correspond exactly to an isoclinic rotation in a Euclidean space of four dimensions. Isospin space resembles the 3-sphere, the curved 3-dimensional space that is the surface of a 4-dimensional ball.

Spinors

A spinor visualized as a vector pointing along the Möbius band, exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.

Spinors are representations of a spin group, which are double covers of the special orthogonal groups. The spin group Spin(4) is the double cover of SO(4), the group of rotations in 4-dimensional Euclidean space. Isoclines, the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4).

Spinors can be viewed as the "square roots" of cross sections of vector bundles; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation).

A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as an isocline can be visualized as a Möbius strip winding twice around the 3-sphere, during which 720° isoclinic rotation the rigid 4-polytope turns itself inside-out twice. Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states.

Isoclinic rotations in the nucleus

Isospin is regarded as a symmetry of the strong interaction under the action of the Lie group SU(2), the two states being the up flavour and down flavour. A 360° isoclinic rotation of a rigid nuclide would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent quarks, by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a rigid polytope undergoing isoclinic rotation.

If the nucleus were a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they were rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities.

Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different hybrid orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination.

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Hybrid isoclinic rotations

The 24-cell has its own characteristic isoclinic rotations in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow helical geodesic curves whose chords form hexagrams, and vertices rotating in square invariant planes follow helical geodesic curves whose chords form octagrams.

In a rigid isoclinic rotation, all the great circle polygons move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described above (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation.

A kinematic isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A hybrid isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell.

As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a valid hybrid rotation.

To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The 18 great squares occur as three sets of 6 orthogonal great squares, each forming a 16-cell. The three 16-cells are completely disjoint and Clifford parallel: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length √2). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell. The two great triangles inscribed in each great hexagon (occupying its alternate vertices, with edges that are its √3 chords) have one vertex in each 16-cell. Thus each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells. Isoclinic rotations take the elements of the 4-polytope to congruent Clifford parallel elements elsewhere in the 4-polytope. The square rotations do this locally, confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act globally within the entire 24-cell: for example, they take great squares to other great squares in different 16-cells. The chords of the square rotations bind the 16-cells together internally, and the chords of the hexagonal rotations bind the three 16-cells together.

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Color

When the existence of quarks was suspected in 1964, Greenberg introduced the notion of color charge to explain how quarks could coexist inside some hadrons in otherwise identical quantum states without violating the Pauli exclusion principle. The modern concept of color charge completely commuting with all other charges and providing the strong force charge was articulated in 1973, by William Bardeen, Harald Fritzsch, and Murray Gell-Mann.

Color charge is not electric charge; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into Clifford parallel subspaces.

The three different colors of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere). Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different sequences of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the only thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline.

If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a rigidly rotating 16-cell, or with a single distinct square rotation.

Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +2/3 in an up quark, and −1/3 in a down quark. One way to do that would be to give the three distinct quarks moving vertices of 1/3 charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left).

Minimally, an up quark requires two moving vertices (of the up+right chirality). Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4.

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Nucleons

Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975. It is the 5-cell. Think of it as an NMR image of the 4-dimensional proton in projection to the plane.

The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the n-body problem, but one is known: the central configuration of n bodies in a space of dimension n-1. A central configuration is a system of point masses with the property that each mass is pulled by the combined attractive force of the system directly towards the center of mass, with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular tetrahedron, five at the vertices of a regular 5-cell, or more generally n masses at the vertices of a regular simplex produces a central configuration even when the masses are not equal. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits.

Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this beta minus decay to occur. The minimal neutron of 4 moving vertices described above cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration. A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration).

It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −1/3, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be.

A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the isoclinic rotations characteristic of the regular 5-cell) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge. In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together.

The most direct geometric relationship among stationary regular 4-polytopes is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which uniquely embodies all the containment relationships among all the regular 4-polytopes and their elements.

If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.

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Nuclides

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Quantum phenomena

The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of d mutually orthogonal lines are white.

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Motion

What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position.

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Galilean relativity in a space of four orthogonal dimensions

Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space $R^{4}$, spherical 4-space $S^{4}$, or any orthogonal 4-manifold.

Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light.

Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions

Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are). The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries.

General relativity is just Galilean relativity in a general space of four orthogonal dimensions

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Physics

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Thoreau's spherical relativity

Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own physical location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: "The universe is a sphere whose center is wherever there is intelligence."

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Conclusions

Spherical relativity

We began our inquiry by wondering why physical space should be limited to just three dimensions (why three). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space $R^{n}$ and in its corresponding spherical space $S^{n}$.

The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a spherical 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes.

One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating.

Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells, but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame.

Nuclear geometry

We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below.

Atomic geometry

I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper:

Atomic geometry

The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy.

flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it

Molecular geometry

Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point.

In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere.

Cosmology

Solar systems

Stars

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The Kepler problem

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Galaxies

The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity c. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ahead of the smaller masses orbiting around it (such as its planets), or is it behind them, as they fly through 4-space on their Clifford parallel trajectories? The answer is both of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere.

What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of.

Revolutions

The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all.

In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe.

When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space.

When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region face on, not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" so they appear to be densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side).

The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity $c$, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity $c$. The view from our solar system does not suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time.

These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since.

Origins of the theory

Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a gedankenexperiment in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.

The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the Minkowski spacetime geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician Ludwig Schläfli around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of Euclidean geometry to spaces of any number of dimensions. He coined the general term polyscheme to mean geometric forms of any number of dimensions, including two-dimensional polygons, three-dimensional polyhedra, four dimensional polychora, and so on, and in the process he discovered all the regular polyschemes that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five Platonic solids in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today Alicia Boole Stott's word polytope is the commonly used term for polyscheme.

Boundaries

Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the polyschemes Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there is a boundary between three and four dimensions. Why can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have three dimensions, and not four, or five, or the n dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of no thickness, a mere abstraction with no physical power to separate, be a more suitable explanation?

The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."

I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

BEECH

Where my imaginary line

Bends square in woods, an iron spine

And pile of real rocks have been founded.

And off this corner in the wild,

Where these are driven in and piled,

One tree, by being deeply wounded,

Has been impressed as Witness Tree

And made commit to memory

My proof of being not unbounded.

Thus truth's established and borne out,

Though circumstanced with dark and doubt—

Though by a world of doubt surrounded.

—The Moodie Forester

Sequence of regular 4-polytopes

Sequence of 8 regular 4-polytopes of radius √2
Symmetry group	A4	B4			F4	H4
Name	5-cell Hyper-tetrahedron 5-point	16-cell Hyper-octahedron 8-point	11-cell Hyper-buckyball 11-point	8-cell Hyper-cube 16-point	24-cell Hyper-cuboctahedron 24-point	600-cell Hyper-icosahedron 120-point	137-cell Hyper-triacontahedron 137-point	120-cell Hyper-dodecahedron 600-point
Schläfli symbol	{3, 3, 3}	{3, 3, 4}	{5/2, 5, 3}	{4, 3, 3}	{3, 4, 3}	{3, 3, 5}	{5/2, 3, 3}	{5, 3, 3}
Coxeter mirrors
Mirror dihedrals	𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2		𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/.. 𝝅/2 𝝅/2 𝝅/2	𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2
Graph
Vertices	5 tetrahedral	8 octahedral	11 dodecahedral	16 tetrahedral	24 cubical	120 icosahedral	137 dodecahedral	600 tetrahedral
Edges	10 triangular	24 square	55 triangular	32 triangular	96 triangular	720 pentagonal	2760 triangular	1200 triangular
Faces	10 triangles	32 triangles	55 triangles	24 squares	96 triangles	1200 triangles	2055 golden rhombi	720 pentagons
Cells	5 {3, 3}	16 {3, 3}	11 {5/2, 3}	8 {4, 3}	24 {3, 4}	600 {3, 3}	137 {5/2, 5}	120 {5, 3}
Tori	5 {3, 3}	8 {3, 3} x 2	11 {5/2, 5}	4 {4, 3} x 2	6 {3, 4} x 4	30 {3, 3} x 20	11 137 {5/2, 3}	10 {5, 3} x 12
Inscribed	120 in 120-cell 96 in 137-cell	2 5-cells 675 in 120-cell	5 16-cells 6 5-cells	2 16-cells 12 5-cells	3 8-cells 3 16-cells	25 24-cells 75 8-cells	9 600-cells 96 24-cells	11 137-cells 10 600-cells
Pentads	1	2	6	12	16	48	96	120
Hexads		2	5	12	16	200	480	600
Heptads			4		4	11	55	120
Great polygons		2 squares x 3	1 11-gon x 1	4 rectangles x 4	4 hexagons x 4	12 decagons x 6	11 11-gons x 11	100 irregular hexagons x 4
Petrie polygons	1 pentagon x 2	1 octagon x 3		2 octagons x 4	2 dodecagons x 4	4 30-gons x 6	11 {11/3}-grams x 11	20 30-gons x 4
Long radius	${\sqrt {2}}$	${\sqrt {2}}$		${\sqrt {2}}$	${\sqrt {2}}$	${\sqrt {2}}$	${\sqrt {2}}$	${\sqrt {2}}$
Edge length	${\sqrt {5}}\approx 2.236$	$2$		${\sqrt {2}}\approx 1.414$	${\sqrt {2}}\approx 1.414$	${\sqrt {4-2\phi }}\approx 0.874$	${\sqrt {5}}\approx 2.236$	$2-\phi \approx 0.382$
Short radius	${\tfrac {\sqrt {2}}{4}}\approx 0.354$	${\tfrac {\sqrt {2}}{2}}\approx 0.707$		${\tfrac {\sqrt {2}}{2}}\approx 0.707$	$1$	${\sqrt {\tfrac {\phi ^{4}}{4}}}\approx 1.309$	${\sqrt {\tfrac {\phi ^{4}}{4}}}\approx 1.309$	${\sqrt {\tfrac {\phi ^{4}}{4}}}\approx 1.309$
Area	$10\left({\tfrac {5{\sqrt {3}}}{4}}\right)\approx 21.651$	$32\left({\sqrt {3}}\right)\approx 55.425$		$48$	$96\left({\sqrt {\tfrac {3}{4}}}\right)\approx 83.138$	$1200\left({\tfrac {2{\sqrt {3}}}{4\phi ^{2}}}\right)\approx 396.95$		$720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{4\phi ^{4}}}\right)\approx 180.73$
Volume	$5\left({\tfrac {5{\sqrt {10}}}{12}}\right)\approx 6.588$	$16\left({\tfrac {2{\sqrt {2}}}{3}}\right)\approx 15.085$		$8{\sqrt {8}}\approx 22.627$	$24\left({\tfrac {4}{3}}\right)=32$	$600\left({\tfrac {4}{12\phi ^{3}}}\right)\approx 47.214$		$120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}}}\right)\approx 51.246$
4-Content	${\tfrac {\sqrt {5}}{24}}\left({\sqrt {5}}\right)^{4}\approx 2.329$	${\tfrac {8}{3}}\approx 2.666$		$4$	$8$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 15.451$		${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 16.770$

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