# Heron’s Triangles and Resonance Decays in Lobachevsky Velocity Space

Valeriy Pavlovich Khen, Aleksei Valerievich Khen

The decay of resonance in the Lobachevsky velocity space is represented by an isosceles triangle inscribed in a horocycle. Near the decay triangles of scalar, strange mesons and Δ, N baryons, there are Heron’s triangles, for which are equal to integers: a) the length of the arc of the oricycle subtending the base, by the cotangent of half the angle at the vertex opposite the base; b) the hyperbolic cosines of the lengths of the lateral sides and the lengths of the bases.

© Valeriy Pavlovich Khen, 2025

© Aleksei Valerievich Khen, 2025


ISBN 978-5-0065-4034-7

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## Abstract

The ends of the velocity vectors of the decay resonance particles are represented by material points of the velocity pace located inside a sphere of radius C (C is the speed of light, the points are assigned the rest masses of the decay particles). The Lorentz group defines in the velocity space the Lobachevsky geometry of curvature k = -1/ C2. A pair of material points of 2-part resonance decay can be connected by a straight line segment and an arc of zero curvature line, called an oricycle [7]. Archimedes’ laws of levers define a 3rd velocity point “m” on the oricycle arc, in which the role of the lever forces is played by the rest masses of the decay particles, and the lever arms are equal to the oricycle arcs. Thus, to the two points with rest masses mπ, a 3-rd velocity point “m” with an additive mass ( mπ+ mπ) is added. Connecting 3 points with straight line segments, we obtain an isosceles triangle Δπmπ of the resonance decay inscribed in the oricycle. In the triangle Δπmπ the golden ratio is discovered and the Stewart and Bretschneider theorems are fulfilled on the arcs of the oricycle (see Appendix). The length of the base of the triangle Δπmπ of the resonance decay is determined by its effective mass.

Near the decay triangles of scalar, strange mesons and Δ, N baryons, satellite triangles with integer values of characteristics were discovered. Namely, for the satellite triangles, the function – the product of the length of the arc of the oricycle subtending their bases, multiplied by the cotangent of half the angle at the vertex opposite the base, takes on integer values. The function is called the oricyclic cotangent of a triangle (OCT) (see Tables 1—4). From Tables 1—4 it is clear that in addition to integer values of OCT, the sum of the hyperbolic cosines of the side lengths and the hyperbolic cosines of the base lengths of the satellite triangles are integers. Therefore, the satellite triangles are called Heron’s resonant triangles.Also in Tables 1—4 are given the obtained multiples of ½ values of the cosines of the angles between the tangent to the velocity point of the base of the Heron’s resonant triangles and the tangent to the velocity point of the additive mass (the values of the cosines > 1 mean that these tangents intersect outside the Absolute and the hyperbolic cosines of the lengths of the segments corresponding to such an angle are calculated).

Thus, the Heron’s resonant decay triangles determine the rigid structure of integers on the infinite radius of the oricycle. When moving along it to the center of the oricycle, new Heron’s resonant triangles with large integer values of OCT can be found. New scalar, strange mesons and Δ, N baryons with large effective masses can be located near them. The formation of Heron’s resonant triangles must be somehow related to the gravitational constant of free fall.

All rights reserved. Certificate No. 53172 dated 31.12.2024. Republic of Kazakhstan.

## Introduction

The ends of the velocity vectors of the decay resonance particles are represented by material points of the velocity space located inside a sphere of radius C (C is the speed of light, the points are assigned the rest masses of the decay particles) [2, 9].

The Lorentz group defines in the velocity space the Lobachevsky geometry of curvature k = -1/ C2 [1—2]. Material points inside a sphere of radius C represent the Beltrami model of hyperbolic Lobachevsky velocity space (HLVS) [2,7].


Fig_1. Separate ellipses of decay oricycles of ƒ0 (500), ρ(775), ƒ0 (980) mesons, combined into one oricycle with the center

at the point “C0“ of the Absolute.


In the Beltrami model the straight lines and planes represent the straight lines and planes of HLVS [7]. The sphere of radius C, called the Absolute, represents infinitely distant points of HLVS.

A pair of material points of 2-part resonance decay in HLVS can be connected by a straight line segment and an arc of zero curvature line, called the oricycle [7,10]. All oricycles are congruent, just as straight lines in Euclidean space are congruent. In Fig_1, in the plane “π”, “π”, “B” (“B” is the velocity of the beam particle B), individual ellipses of oricycles of scalar mesons, whose centers “C0” lie on the Absolute, combined into one oricycle with a common center “C0”.

Archimedes’ laws of levers define a 3rd velocity point “m” on the oricycle arc, in which the role of the lever forces is played by the rest masses of the decay particles, and the lever arms are equal to the oricycle arcs. Thus, to the two material points “π”, “π” with rest masses mπ, a 3-rd velocity point “m” with an additive mass (m