User:Dc.samizdat/Rotations
Abstract: The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. 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
[edit | edit source]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.[1]
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
[edit | edit source]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 partially synchronized 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.
Rotations
[edit | edit source]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 precisely detailed description enabling the reader to visualize it runs to many pages and illustrations, and requires many more 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 much 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.[a] 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
[edit | edit source]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 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.[2]:129–130 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, well before the 1960s development of the quark model, by Werner Heisenberg,[3] 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.[4] 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). It does seem to correspond exactly to an isoclinic rotation in a Euclidean space of four dimensions. Isospin space is the 3-sphere, the curved 3-dimensional space that is the surface of a 4-dimensional ball.
Spinors
[edit | edit source]![](http://178.128.105.246/content-http-upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Spinor_on_the_circle.png/330px-Spinor_on_the_circle.png)
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
[edit | edit source]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
[edit | edit source]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 rotations 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,[b] each forming a 16-cell. The three 16-cells are completely disjoint[c] and Clifford parallel: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length √2).[d] The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.[e] 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.[g] 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
[edit | edit source]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.[5][6]
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).[h] 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. If the neutron is a (8-point) 16-cell, either possibility might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter possibility makes sense, because fibrations (distinct isoclinic left (right) 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: each fibration . Second, when one considers the 3 isoclines of each 4-polytope, in each of the three fibrations two isoclines carry the left and right rotations, 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, 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).[j] 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
[edit | edit source]![](http://178.128.105.246/content-http-upload.wikimedia.org/wikipedia/commons/thumb/1/10/Symmetrical_5-set_Venn_diagram.svg/220px-Symmetrical_5-set_Venn_diagram.svg.png)
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 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.[k] 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 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.[l] In particular, they give no account of 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 the 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.[m]
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Nuclides
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Quantum phenomena
[edit | edit source]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.[n]
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Motion
[edit | edit source]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
[edit | edit source]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 , spherical 4-space , 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
[edit | edit source]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).[8] 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 they are all 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
[edit | edit source]Every observer properly sees himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points in it, including his own physical location which is one of those points moving at speed , on a 4-vector which is distinguished to him, but is 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.[q] 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."[11]
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Conclusions
[edit | edit source]Spherical relativity
[edit | edit source]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". The laws of physics may be considered to operate in any flat Euclidean space and in its corresponding spherical space .
One phenomenon which resists explaination 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 is 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 4-simplex and 4-orthoplex polytopes (the 5-cell proton and the 16-cell neutron). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,[12] but it is only directly accessible (absent a 120-cell) in a five-dimensional reference frame.
Nuclear geometry
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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 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, and the 3-spherical membrane in which we find ourselves living must be one of many 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 almost certainly not all located on our own 3-spherical membrane. Perhaps the galaxies are single roughly spherical 3-membranes, each with a single big bang origin point in 4-space and time, and smaller objects within them all lie on that same 3-spherical membrane. But other cosmological objects of galactic size must be of other origins at the centers of their own 3-spheres, and we have few reasons to suppose that their collective origins should be singular either.
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
[edit | edit source]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 relativity, contemporaneous with the teen-aged Coxeter's (quoted below). He did this as a gedankenexperiment in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe. But when in his 1921 Princeton lecture he invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions"[13], he was careful to note 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, when 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 same ground, and competing terms were coined; today Alicia Boole Stott's word polytope is the commonly used term for polyscheme.[r]
Boundaries
[edit | edit source]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.[14]
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 space the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are confined within the curved boundary itself. Again, our unconscious provincial concept is that there is nothing else: 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."[15]
I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. But others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter, 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[16]
Sequence of regular 4-polytopes
[edit | edit source]Sequence of 8 regular 4-polytopes of radius √2 | ||||||||
---|---|---|---|---|---|---|---|---|
Symmetry group | A4 | B4 | F4 | H4 | ||||
Name | 5-cell Hyper-tetrahedron |
16-cell Hyper-octahedron |
11-cell Hyper-buckyball |
8-cell Hyper-cube |
24-cell Hyper-cuboctahedron |
600-cell Hyper-icosahedron |
137-cell Hyper-triacontahedron |
120-cell Hyper-dodecahedron |
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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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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 | ![]() |
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Vertices[s] | 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[t] | 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 | ||||||||
Edge length[u] | ||||||||
Short radius | ||||||||
Area | ||||||||
Volume | ||||||||
4-Content |
Notes
[edit | edit source]- ↑ 1.0 1.1 Departing from any vertex V0 in the original great hexagon plane of isoclinic rotation P0, the first vertex reached V1 is 120 degrees away along a √3 chord lying in a different hexagonal plane P1. P1 is inclined to P0 at a 60° angle.[as] The second vertex reached V2 is 120 degrees beyond V1 along a second √3 chord lying in another hexagonal plane P2 that is Clifford parallel to P0.[au] (Notice that V1 lies in both intersecting planes P1 and P2, as V0 lies in both P0 and P1. But P0 and P2 have no vertices in common; they do not intersect.) The third vertex reached V3 is 120 degrees beyond V2 along a third √3 chord lying in another hexagonal plane P3 that is Clifford parallel to P1. The three √3 chords lie in different 8-cells.[ae] V0 to V3 is a 360° isoclinic rotation.
- ↑ 2.0 2.1 2.2 2.3 2.4 In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is completely orthogonal to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.
- ↑ 3.0 3.1 3.2 3.3 Polytopes are completely disjoint if all their element sets are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.
- ↑ 4.0 4.1 Visualize the three 16-cells inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes;[b] the other two are rotated 60° isoclinically to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.[y] The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are √2, each vertex of the compound of three 16-cells is √1 away from its 8 surrounding vertices in other 16-cells. Now visualize those √1 distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The √1 edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. Four hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.[z] The hexagons are not perpendicular to each other, or to the 16-cells' perpendicular square central planes.[e] The left and right 16-cells form a tesseract.[f] Two 16-cells have vertex-pairs which are one √1 edge (one hexagon edge) apart. But a simple rotation of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell can be taken to another 16-cell by a 60° isoclinic rotation, because an isoclinic rotation is 3-sphere symmetric: four Clifford parallel hexagonal planes rotate together, but in four different rotational directions,[x] taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a diagonal rotation by 60° in two completely orthogonal directions at once,[af] the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: two √1 hexagon edges (or one √3 hexagon chord) apart, not one √1 edge (60°) apart as in a simple rotation.[w] By the chiral diagonal nature of isoclinic rotations, the 16-cell cannot reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell beyond it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation will take every 16-cell to another 16-cell: a 60° right isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the left 16-cell, and a 60° left isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the right 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)
- ↑ 5.0 5.1 5.2 The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only one of the 4 coordinate system axes.[ad] The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of integer coordinate vertices (one of the four coordinate axes), and two opposite pairs of half-integer coordinate vertices (not coordinate axes). For example:
( 0, 0, 1, 0)
( 1/2,–1/2, 1/2,–1/2) ( 1/2, 1/2, 1/2, 1/2)
(–1/2,–1/2,–1/2,–1/2) (–1/2, 1/2,–1/2, 1/2)
( 0, 0,–1, 0)
is a hexagon on the y axis. Unlike the √2 squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell. - ↑ 6.0 6.1 Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional hypercube (a tesseract or 8-cell), in dimensional analogy to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.[ae] The tesseracts share vertices, but the 16-cells are completely disjoint.[c]
- ↑ There are four different ways (four different fibrations of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices √3 apart: there are 32 distinct linking triangles. Each pair of 16-cells forms a tesseract (8-cell).[f] Each great triangle has one √3 edge in each tesseract, so it is also a ring linking the three tesseracts.
- ↑ The 8 vertices of each disjoint 16-cell constitute an independent orthonormal basis for a coordinate reference frame.
- ↑ There is only one octagram orbit of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at every vertex.
- ↑ Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.[i] However, the theory of color confinement may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.
- ↑ Although protons are very stable, during stellar nucleosynthesis two H1 protons are fused into an H2 nucleus consisting of a proton and a neutron. This beta plus "decay" of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.
- ↑ Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.
- ↑ The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The isoclinic rotation characteristic of the 16-cell maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) may be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, i.e. they form a rigid 5-vertex polytope undergoing some kind of rotation. Notice that this 5-vertex polytope is not a 5-cell, since 4 of its vertices are coplanar; it is not a 4-polytope but merely a polyhedron, a square pyramid.
- ↑ "The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."[7]
- ↑ Cayley showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a rotation in 4-dimensional Euclidean space.
- ↑ Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for one of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of one rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.
- ↑ Let Q denote a rotation, R a reflection, T a translation, and let Qq Rr T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q2 is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
Qq Rr
where 2q + r ≤ n, the number of dimensions. Transformations involving a translation are expressible as
Qq Rr T
where 2q + r + 1 ≤ n.
For n = 4 in particular, every displacement is either a double rotation Q2, or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the Galilean principle of relativity, every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q2 in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q2. By the same principle, we can view any QT or Q2 as an isoclinic (equi-angled) Q2 by appropriate choice of reference frame.[o] That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.[p]] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.[9] - ↑ Today Schläfli's original polyscheme, with its echo of schema as in the configurations of information structures, seems even more fitting in its generality than polytope -- perhaps analogously as information software (programming) is even more general than information hardware (computers).
- ↑ Cite error: Invalid
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- ↑ 22.0 22.1 22.2 Clifford parallels are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.[20] A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the 3-sphere.[21] Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.[b] Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.[ak] Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a Hopf link.
- ↑ 23.0 23.1 23.2 23.3 23.4 23.5 In an isoclinic rotation, each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a 4-dimensional diagonal. The point is displaced a total W:Pythagorean distance equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,[am] all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex √3 (120°) away, moving √3/4 in four orthogonal coordinate directions.
- ↑ 24.0 24.1 In a Clifford displacement, also known as an isoclinic rotation, all the Clifford parallel[v] invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted sideways by that same angle. A Clifford displacement is 4-dimensionally diagonal.[w] Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.
- ↑ 25.0 25.1 25.2 Eight √1 edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure[ab] and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two √1-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a cubic pyramid. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.[ac]
- ↑ 26.0 26.1 It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the cuboctahedron. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the edges around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].[y]
- ↑ 27.0 27.1 The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and tesseract, the three-dimensional cuboctahedron, and the two-dimensional hexagon. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.
- ↑ The vertex figure is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a full size vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".[17] That is what serves the illustrative purpose here.
- ↑ The vertex cubic pyramid is not actually radially equilateral,[aa] because the edges radiating from its apex are not actually its radii: the apex of the cubic pyramid is not actually its center, just one of its vertices.
- ↑ Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically[w] with respect to each other (so their corresponding vertices are 120° = √3 apart). A 16-cell is an orthonormal basis for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only one axis which is a coordinate system axis.
- ↑ 31.0 31.1 31.2 The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are √3 (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four √3 chords (its long diagonals). The 8-cells are not completely disjoint[c] (they share vertices), but each cube and each √3 chord belongs to just one 8-cell. The √3 chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.
- ↑ 32.0 32.1 A point under isoclinic rotation traverses the diagonal[w] straight line of a single isoclinic geodesic, reaching its destination directly, instead of the bent line of two successive simple geodesics. A geodesic is the shortest path through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do not lie in a single plane; they are 4-dimensional spirals rather than simple 2-dimensional circles.[ag] But they are not like 3-dimensional screw threads either, because they form a closed loop like any circle (after two revolutions). Isoclinic geodesics are 4-dimensional great circles, and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.[ah] These isoclines are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere[ai] they always occur in chiral pairs and form a pair of Villarceau circles on the Clifford torus,[aj] the paths of the left and the right isoclinic rotation. They are helices bent into a Möbius loop in the fourth dimension, taking a diagonal winding route twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's skew polygon.
- ↑ 33.0 33.1 In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be invariant because the points in each stay in the plane as the plane moves, tilting sideways by the same angle that the other plane rotates.
- ↑ Isoclinic geodesics are 4-dimensional great circles in the sense that they are 1-dimensional geodesic lines that curve in 4-space in two completely orthogonal planes at once. They should not be confused with great 2-spheres,[18] which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).
- ↑ All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.
- ↑ Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a Hopf link called the {1,1} torus knot[19] in which each of the two linked circles traverses all four dimensions.
- ↑ 37.0 37.1 37.2 Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal[al] to only one of them.[an] Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).
- ↑ 38.0 38.1 38.2 38.3 Two flat planes A and B of a Euclidean space of four dimensions are called completely orthogonal if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.[b]
- ↑ 39.0 39.1 In the 24-cell each great square plane is completely orthogonal[al] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great digon plane.
- ↑ 40.0 40.1 40.2 In the 16-cell the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically[w] with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.[ak]) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.
- ↑ To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w=0, z=0) shares no axis with the wz central plane (where x=0, y=0). The xy plane exists at only a single instant in time (w=0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).
- ↑ Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) they can intersect in a single point[ao] (and they must, if they are completely orthogonal).[al]
- ↑ 43.0 43.1 If the Pythagorean distance between any two vertices is √1, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is √2, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90o bend in it as the path through the center). If their Pythagorean distance is √3, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60o bend, or as a straight line with one 60o bend in it through the center). Finally, if their Pythagorean distance is √4, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).
- ↑ 44.0 44.1 44.2 Two angles are required to fix the relative positions of two planes in 4-space.[22] Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in both angles. Great squares in different hyperplanes are 90 degrees apart in both angles (completely orthogonal)[al] or 60 degrees apart in both angles.[an] Planes which are separated by two equal angles are called isoclinic. Planes which are isoclinic have Clifford parallel great circles.[v] A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle and a 60 degree angle.
- ↑ P0 and P1 lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.[ar]
- ↑ V0 and V2 are two √3 chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than one √3 chord, unless they are antipodal vertices √4 apart.[aq] V0 and V2 are one √3 chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their adjacent vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).
- ↑ P0 and P2 are 60° apart in both angles of separation.[ar] Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V0 and V2 are two √3 chords apart[at], P0 and P2 are just one √1 edge apart (at every pair of nearest vertices).
Citations
[edit | edit source]- ↑ Conway, Burgiel & Goodman-Strauss 2008.
- ↑ Griffiths, David J. (2008). Introduction to Elementary Particles (2nd revised ed.). WILEY-VCH. ISBN 978-3-527-40601-2.
- ↑ Heisenberg, W. (1932). "Über den Bau der Atomkerne". Zeitschrift für Physik 77 (1–2): 1–11. doi:10.1007/BF01342433.
- ↑ Wigner, E. (1937). "On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei". Physical Review 51 (2): 106–119. doi:10.1103/PhysRev.51.106.
- ↑ Bardeen, W.; Fritzsch, H.; Gell-Mann, M. (1973). "Light cone current algebra, π0 decay, and e+ e− annihilation". In Gatto, R. (ed.). Scale and conformal symmetry in hadron physics. John Wiley & Sons. p. 139. arXiv:hep-ph/0211388. Bibcode:2002hep.ph...11388B. ISBN 0-471-29292-3.
- ↑ Fritzsch, H.; Gell-Mann, M.; Leutwyler, H. (1973). "Advantages of the color octet gluon picture". Physics Letters B 47 (4): 365. doi:10.1016/0370-2693(73)90625-4.
- ↑ Waegell & Aravind 2009, 2. The Bell-Kochen-Specker (BKS) theorem.
- ↑ Yamashita 2023.
- ↑ Coxeter 1973, pp. 217-218, §12.2 Congruent transformations.
- ↑ Coxeter 1973, pp. 141-144, §7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."
- ↑ Thoreau 1849, p. 349; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous with Ludwig Schlafli's pioneering work enumerating the complete set of regular polytopes in any number of dimensions.[10]]
- ↑ Christie 2024.
- ↑ Einstein, Albert (1923). The Meaning of Relativity. Princeton University Press. pp. 110-111. http://www.gutenberg.org/ebooks/36276.
- ↑ Carlo Rovelli. Seven Brief Lessons on Physics.
- ↑ Coxeter, Donald (February 1923), Dimensional Analogy, Coxeter Fonds, University of Toronto Archives
- ↑ Frost, Robert (1942). A Witness Tree. The Poetry of Robert Frost (1969 ed.). Holt, Rinehart and Winston.
- ↑ Stillwell 2001, p. 17.
- ↑ Stillwell 2001, p. 24.
- ↑ Dorst 2019, p. 44, §1. Villarceau Circles; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a Villarceau circle. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a Hopf fibration.... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a Hopf link] rather than as a planar cut [two intersecting circles]."
- ↑ Tyrrell & Semple 1971, pp. 5-6, §3. Clifford's original definition of parallelism.
- ↑ Kim & Rote 2016, pp. 8-10, Relations to Clifford Parallelism.
- ↑ Kim & Rote 2016, p. 7, §6 Angles between two Planes in 4-Space; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, k angles are defined between k-dimensional subspaces.)"
- ↑ Mamone, Pileio & Levitt 2010, pp. 1438-1439, §4.5 Regular Convex 4-Polytopes; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹4.
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