The Dimensional Gate – 3D rotational shadows of E8 Polytope
&gt;&quot;A demonstration of the infinite number of three dimensional shadows of the infinite number of rotations of the E8 Polytope in the eighth dimension.&quot;<p>PDS: Let&#x27;s break all of this down:<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Polytope" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Polytope</a><p>&gt;&quot;In elementary geometry, a polytope is a geometric object with &quot;flat&quot; 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.&quot;<p>[...]<p>Polytopes in lower numbers of dimensions have standard names:<p>Dimension
of polytope Description[6]<p>−1 Nullitope<p>0 Monon<p>1 Dion<p>2 Polygon<p>3 Polyhedron<p>4 Polychoron<p>&gt;<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Coxeter_element" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Coxeter_element</a><p>&gt;&quot;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]&quot;<p>Definitions<p>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.<p>There are many different ways to define the Coxeter number h of an irreducible root system.<p>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.<p>The Coxeter number is the order of any Coxeter element;.
The Coxeter number is 2m&#x2F;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.&quot;<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Coxeter_element#Coxeter_plane" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Coxeter_element#Coxeter_plane</a><p>&gt;&quot;For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π&#x2F;h. This is called the Coxeter plane[6] and is the plane on which P has eigenvalues e2πi&#x2F;h and e−2πi&#x2F;h = e2πi(h−1)&#x2F;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]<p>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.<p>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.<p>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&#x2F;h[Ch×Ch][10] (John H. Conway), (C2h&#x2F;C1;C2h&#x2F;C1) (#1&#x27;, Patrick du Val (1964)[11]), order h.<p>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.<p>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.<p>Orthographic projection<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Orthographic_projection" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Orthographic_projection</a><p>&gt;&quot;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.&quot;<p>PDS: I&#x27;m guessing Orthographic Projections -- can also be n-dimensional -- from dimension X to dimension Y -- as opposed to just 3D to 2D...<p>Also, note Orthographic Projections relationship with Mathematical Matrixes&#x2F;Matrices (and Computer Graphics)...<p>E8 Polytope<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;E8_polytope" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;E8_polytope</a><p>&gt;&quot;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.&quot;<p>These polytopes can be visualized as symmetric orthographic projections in Coxeter planes of the E8 Coxeter group, and other subgroups.<p>Graphs<p>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.<p>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.&quot;<p>Keywords: Future Sci-Fi Gate Technology (?), Hologram &#x2F; Holographic Projection, Superposition, Platonic Solids, Structure Of Matter, Structure Of Antimatter (Conjugate), N-Dimensional Symmetry, Polyhedra, Crystallography, Higher Dimensional Shadows, Warp Drive (?), Portal Technology (?), &quot;Any sufficiently advanced technology is indistinguishable from magic&quot; etc., etc.