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PapersGrego — Gergely Földvári
Twelve scientific papers in geometry and number theory, related to games, chemistry, music, language, sports, architecture, art and religion, written between 2016 and 2026. Nearly all of them meet at the same number: the cube has 26 characteristic elements, 26 is the only integer wedged between two perfect powers, 26 is iron, 26 are the letters of the English alphabet and 26 is the magic sum of the sieve model, to just name a few.
250 pages · twelve DOIs ·
CC BY 4.0
A polyhedral sieve model for medical diagnostics
The magic hexagram mounted on Escher's solid, the stellated rhombic dodecahedron. Three sieves — equilibrium, unambiguity, reliability — reduce a factorial numbering larger than the atom count of the Solar System to a bounded landscape whose full census is 6,223,974 pyramid-magic numberings, with one singularity. Put forward as a candidate vocabulary for the diagnosis of rare disease.
26 June 2026100 pages
DOI 10.5281/zenodo.22259867
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A follow up to Star Face
The Star Face census regrouped by the depth of the deepest mask: 56.85% of numberings show one clear reading, 42.49% a two-way ambiguity, 0.66% a three-way and 0.00027% a four-way. What the medical literature holds at each depth, why masking is six different words in one, and why the diagnostic odyssey of rare disease is the only evidence reaching past depth two. Closes with the physician survey that would test the model, set out ready to copy.
8 September 202618 pages
DOI 10.5281/zenodo.22659188
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A new word for an old arithmetic, shown on Platonic and Archimedean solids
Summetry is symmetry that carries arithmetic: where symmetry says which parts of a figure are equivalent, summetry requires them to sum alike, and the shape then fixes what the sum must be. S = mT/r. All five Platonic and all thirteen Archimedean solids carry the same arrangement — twelve things, six groups of four — and therefore the same constant, 26, through the same 960 numberings. Of the twenty-four arrangements of that shape, exactly eight can carry 26 and two can never be a solid.
8 September 202614 pages
DOI 10.5281/zenodo.22658904
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and how merit is weighed over luck
Three studies of a memory, strategy and lottery game. How many categories the Feast admits; whether two perfect readers of one shuffle differ, one following the loops and one scanning the rows; and where the depth of the game comes from. Built on permutation cycles and the author's own integer sequence.
2 September 202651 pages
DOI 10.5281/zenodo.22260941
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A new class of square dissections, with a smallest identified
Medical and demographic data arrive as whole percentage points, so carry that constraint into geometry: squared squares and rectangles whose element sides all lie between 1 and 100 and hold both extremes. The perfect members are finite — order at most 100, side at most 581 — so the class can be counted. Of 248,169 dissections inside the scale, exactly eight are perfect squared squares; the smallest, 26:274A, is the only one whose median reaches fifty.
7 September 202615 pages
DOI 10.5281/zenodo.22304807
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A new class of constants in geometry
The cube has 26 characteristic elements and the English alphabet has 26 letters. Labelling one with the other gives the English Cube, whose 253 connecting segments have a constant total length — 280.9107 for the unit cube. With the notation, the eight segment classes, and the (MN) sign designed for it.
26 January 202512 pages
DOI 10.5281/zenodo.22259625
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A tetradecahedron from the truncated cube, and the Eternal Flame of the Dohány Street Synagogue
A cube truncated into a solid of 14 faces — 2 squares, 4 hexagons, 8 triangles — then augmented with twelve pyramids: the lantern of the Eternal Flame above the bimah of the Great Synagogue in Budapest. Three specialists in polyhedral geometry each drew it, and none had seen it before.
13 August 20238 pages
DOI 10.5281/zenodo.22261829
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and 26, the Transdimensional Number
Add 26% to the edge of a cube and you about double its volume.
The cube root of 2 is 1.2599210, so the edge of a cube of
double the volume is a little over 26% longer. Rounded to 1.26 the rule can be
done in the head and is accurate to two hundredths of one per cent — and
it is not a fact about cubes. Volume scales as the cube of any length, so 26%
added to any measurement of any shape, from a teapot to a galaxy, about doubles
what it holds.
1 May 20248 pages
DOI 10.5281/zenodo.22282044
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Multimorphism, transdimension and singularity
26 is the only integer directly sandwiched between perfect powers. A fresh approach to that folklore problem, congruent to the case of integer 2 in Pillai's conjecture, introducing coefficial divisibility, the multimorphic integer and the transdimensional number.
17 April 20256 pages
DOI 10.5281/zenodo.22259775
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Cube edge times body- over face-diagonal equals the diameter of the largest inscribed circle
The largest circle a cube will hold is wider than the cube's own edge. Stated as a sentence, needing no square roots: they appear only in the proof. Two extensions follow, to the cube's largest inscribed triangle and square.
19 April 20246 pages
DOI 10.5281/zenodo.22259163
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The major scale and the Hebrew leap-year cycle
The Hebrew calendar inserts a thirteenth month in seven years out of every nineteen, at positions 3, 6, 8, 11, 14, 17 and 19. Count each step of the major scale by the notes it spans — three for a whole step, two for a half — and the scale gives exactly that list. Not a coincidence but a theorem: both are the evenest way to place seven things in a cycle that leaves remainder five. The seven modes are the seven ways of entering it, and the Hebrew cycle turns out to be the major scale in nineteen-tone equal temperament.
19 February 20166 pages
DOI 10.5281/zenodo.22309689
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Versenykimenetel képlet — bilingual edition
How many ways can a race end? Clean finishes give k factorial; dead heats give the Fubini numbers. But a race also loses competitors, and being disqualified after the start is not the same event as never starting — one leaves a split time on the clock and something for an inquiry, the other leaves nothing. Counting all of it gives eleven outcomes for two competitors, 299 for four, and over 112 billion for twelve. That single distinction doubles the count.
10 July 20246 pages
DOI 10.5281/zenodo.22309791
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