3 ms·

The Dimensional Gate – 3D rotational shadows of E8 Polytope

peter_d_sherman 6y ago
>"A demonstration of the infinite number of three dimensional shadows of the infinite number of rotations of the E8 Polytope in the eighth dimension." PDS: Let's break all of this down: https://en.wikipedia.org/wiki/Polytope https://en.wikipedia.org/wiki/Polytope >"In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope." [...] Polytopes in lower numbers of dimensions have standard names: Dimension of polytope Description[6] −1 Nullitope 0 Monon 1 Dion 2 Polygon 3 Polyhedron 4 Polychoron >https://en.wikipedia.org/wiki/Coxeter_element https://en.wikipedia.org/wiki/Coxeter_element >"In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.[1]" Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number h of an irreducible root system. A Coxeter element is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. The Coxeter number is the order of any Coxeter element;. The Coxeter number is 2m/n, where n is the rank, and m is the number of reflections. In the crystallographic case, m is half the number of roots; and 2m+n is the dimension of the corresponding semisimple Lie algebra. If the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi. The Coxeter number is the highest degree of a fundamental invariant of the Coxeter group acting on polynomials." https://en.wikipedia.org/wiki/Coxeter_element#Coxeter_plane https://en.wikipedia.org/wiki/Coxeter_element#Coxeter_plane >"For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h. This is called the Coxeter plane[6] and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi(h−1)/h.[7] This plane was first systematically studied in (Coxeter 1948),[8] and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.[8] The Coxeter plane is often used to draw diagrams of higher-dimensional polytopes and root systems – the vertices and edges of the polytope, or roots (and some edges connecting these) are orthogonally projected onto the Coxeter plane, yielding a Petrie polygon with h-fold rotational symmetry.[9] For root systems, no root maps to zero, corresponding to the Coxeter element not fixing any root or rather axis (not having eigenvalue 1 or −1), so the projections of orbits under w form h-fold circular arrangements[9] and there is an empty center, as in the E8 diagram at above right. For polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids. In three dimensions, the symmetry of a regular polyhedron, {p,q}, with one directed Petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Sh, [2+,h+], order h. Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, Dhd, [2+,h], order 2h. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, [h], order 2h. In four dimensions, the symmetry of a regular polychoron, {p,q,r}, with one directed Petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h[Ch×Ch][10] (John H. Conway), (C2h/C1;C2h/C1) (#1', Patrick du Val (1964)[11]), order h. In five dimensions, the symmetry of a regular 5-polytope, {p,q,r,s}, with one directed Petrie polygon marked, is represented by the composite of 5 reflections. In dimensions 6 to 8 there are 3 exceptional Coxeter groups; one uniform polytope from each dimension represents the roots of the exceptional Lie groups En. The Coxeter elements are 12, 18 and 30 respectively. Orthographic projection https://en.wikipedia.org/wiki/Orthographic_projection https://en.wikipedia.org/wiki/Orthographic_projection >"Orthographic projection (sometimes referred to as orthogonal projection, used to be called analemma[a]) is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane,[2] resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane." PDS: I'm guessing Orthographic Projections -- can also be n-dimensional -- from dimension X to dimension Y -- as opposed to just 3D to 2D... Also, note Orthographic Projections relationship with Mathematical Matrixes/Matrices (and Computer Graphics)... E8 Polytope https://en.wikipedia.org/wiki/E8_polytope https://en.wikipedia.org/wiki/E8_polytope >"In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of 240, 2160 and 17280 vertices respectively." These polytopes can be visualized as symmetric orthographic projections in Coxeter planes of the E8 Coxeter group, and other subgroups. Graphs Symmetric orthographic projections of these 255 polytopes can be made in the E8, E7, E6, D7, D6, D5, D4, D3, A7, A5 Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry, and E6, E7, E8 have [12], [18], [30] symmetry respectively. In addition there are two other degrees of fundamental invariants, order [20] and [24] for the E8 group that represent Coxeter planes. 11 of these 255 polytopes are each shown in 14 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position." Keywords: Future Sci-Fi Gate Technology (?), Hologram / Holographic Projection, Superposition, Platonic Solids, Structure Of Matter, Structure Of Antimatter (Conjugate), N-Dimensional Symmetry, Polyhedra, Crystallography, Higher Dimensional Shadows, Warp Drive (?), Portal Technology (?), "Any sufficiently advanced technology is indistinguishable from magic" etc., etc.

>"A demonstration of the infinite number of three dimensional shadows of the infinite number of rotations of the E8 Polytope in the eighth dimension." PDS: Let's break all of this down: https://en.wikipedia.org/wiki/Polytope https://en.wikipedia.org/wiki/Polytope >"In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope." [...] Polytopes in lower numbers of dimensions have standard names: Dimension of polytope Description[6] −1 Nullitope 0 Monon 1 Dion 2 Polygon 3 Polyhedron 4 Polychoron >https://en.wikipedia.org/wiki/Coxeter_element https://en.wikipedia.org/wiki/Coxeter_element >"In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.[1]" Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number h of an irreducible root system. A Coxeter element is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. The Coxeter number is the order of any Coxeter element;. The Coxeter number is 2m/n, where n is the rank, and m is the number of reflections. In the crystallographic case, m is half the number of roots; and 2m+n is the dimension of the corresponding semisimple Lie algebra. If the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi. The Coxeter number is the highest degree of a fundamental invariant of the Coxeter group acting on polynomials." https://en.wikipedia.org/wiki/Coxeter_element#Coxeter_plane https://en.wikipedia.org/wiki/Coxeter_element#Coxeter_plane >"For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h. This is called the Coxeter plane[6] and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi(h−1)/h.[7] This plane was first systematically studied in (Coxeter 1948),[8] and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.[8] The Coxeter plane is often used to draw diagrams of higher-dimensional polytopes and root systems – the vertices and edges of the polytope, or roots (and some edges connecting these) are orthogonally projected onto the Coxeter plane, yielding a Petrie polygon with h-fold rotational symmetry.[9] For root systems, no root maps to zero, corresponding to the Coxeter element not fixing any root or rather axis (not having eigenvalue 1 or −1), so the projections of orbits under w form h-fold circular arrangements[9] and there is an empty center, as in the E8 diagram at above right. For polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids. In three dimensions, the symmetry of a regular polyhedron, {p,q}, with one directed Petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Sh, [2+,h+], order h. Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, Dhd, [2+,h], order 2h. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, [h], order 2h. In four dimensions, the symmetry of a regular polychoron, {p,q,r}, with one directed Petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h[Ch×Ch][10] (John H. Conway), (C2h/C1;C2h/C1) (#1', Patrick du Val (1964)[11]), order h. In five dimensions, the symmetry of a regular 5-polytope, {p,q,r,s}, with one directed Petrie polygon marked, is represented by the composite of 5 reflections. In dimensions 6 to 8 there are 3 exceptional Coxeter groups; one uniform polytope from each dimension represents the roots of the exceptional Lie groups En. The Coxeter elements are 12, 18 and 30 respectively. Orthographic projection https://en.wikipedia.org/wiki/Orthographic_projection https://en.wikipedia.org/wiki/Orthographic_projection >"Orthographic projection (sometimes referred to as orthogonal projection, used to be called analemma[a]) is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane,[2] resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane." PDS: I'm guessing Orthographic Projections -- can also be n-dimensional -- from dimension X to dimension Y -- as opposed to just 3D to 2D... Also, note Orthographic Projections relationship with Mathematical Matrixes/Matrices (and Computer Graphics)... E8 Polytope https://en.wikipedia.org/wiki/E8_polytope https://en.wikipedia.org/wiki/E8_polytope >"In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of 240, 2160 and 17280 vertices respectively." These polytopes can be visualized as symmetric orthographic projections in Coxeter planes of the E8 Coxeter group, and other subgroups. Graphs Symmetric orthographic projections of these 255 polytopes can be made in the E8, E7, E6, D7, D6, D5, D4, D3, A7, A5 Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry, and E6, E7, E8 have [12], [18], [30] symmetry respectively. In addition there are two other degrees of fundamental invariants, order [20] and [24] for the E8 group that represent Coxeter planes. 11 of these 255 polytopes are each shown in 14 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position." Keywords: Future Sci-Fi Gate Technology (?), Hologram / Holographic Projection, Superposition, Platonic Solids, Structure Of Matter, Structure Of Antimatter (Conjugate), N-Dimensional Symmetry, Polyhedra, Crystallography, Higher Dimensional Shadows, Warp Drive (?), Portal Technology (?), "Any sufficiently advanced technology is indistinguishable from magic" etc., etc.