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[5] viXra:2505.0189 [pdf] submitted on 2025-05-28 20:30:24
Authors: Miquel Piñol
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We introduce a class of geometric bodies, which we call vertex-edge-combinatorial polytopes, defined by a local structure in which each vertex is connected to a number of edges equal to the dimension of the body, and where any subset of those edges belongs to a face whose dimension equals the subset’s cardinality. These polytopes satisfy an empirical formula for the number of vertices, from which a general combinatorial expression for the number of faces can be deduced. The class includes simplices, hypercubes, and the dodecahedron, and excludes the octahedron, the icosahedron, and any higher-dimensional polytopes derived from them. In some cases where the formula diverges, such as the hexagonal tiling, an infinite regular structure does exist, although this is not always the case.
Category: Geometry
[4] viXra:2505.0102 [pdf] submitted on 2025-05-16 02:56:15
Authors: Andrey V. Voron
Comments: 15 Pages.
The possibility of constructing Platonic solids from structural elements is shown — Kepler triangles (ratio of legs 1:√1.618..) and Fibonacci (ratio of legs 1:1.618...) — provided that the area of these elements remains unchanged. The number of elements (or pairs of elements) that make up the structure of the "tetrahedron", "octahedron", "cube" increases, thus, by two times, and the "icosahedron" — by five times in relation to the number of elements of the tetrahedron, while the indicator "area of all structural elements of the figure" and radius (r=3) remain unchanged inscribed in the Platonic bodies of the sphere. In addition, the area of the structural elements of two dodecahedra (S=√959325) is equal to the area of the structural elements of any 5 Platonic solids, for example, 5 tetrahedra (or octahedra, cubes, icosahedra) (S=√38373). The possibility shown is in accordance with the text of Plato's work Timaeus, according to which Platonic bodies can "transform into each other...".
Category: Geometry
[3] viXra:2505.0092 [pdf] submitted on 2025-05-14 20:09:10
Authors: Andrey V. Voron
Comments: 2 Pages. (Note by viXra Admin: Please don't use all CAPS in the article title)
The article shows the possibility of using a right triangle and the logic of the Pythagorean theorem to find the areas of regular two-dimensional and three-dimensional geometric shapes (in particular, for Platonic solids).
Category: Geometry
[2] viXra:2505.0091 [pdf] submitted on 2025-05-14 20:08:19
Authors: Andrey V. Voron
Comments: 3 Pages. (Note by viXra Admin: Please don't use all CAPS in the article title)
The article shows the possibility (using an acute-angled triangle, the logic of the Pythagorean theorem) of finding the volume of regular three-dimensional geometric shapes based on the mathematical equation c=3√(a3+b3). A number of theorems have been formulated that complement the Pythagorean theorem.
Category: Geometry
[1] viXra:2505.0066 [pdf] submitted on 2025-05-10 09:56:06
Authors: Shanzhong Zou
Comments: 4 Pages.
This paper constructs a "three-dimensional complex coordinate system" and proposes that the non-trivial zeros of the Riemann Zeta function lie on demarcation lines within the critical region. By infinitely projecting the singular point s=1 between complex planes in the 3D complex space and the standard complex plane, we derive regions where non-trivial zeros cannot exist. The boundaries of these regions (including Re(s) =1/2) are identified as potential loci for zeros.
Category: Geometry
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