# Bulletin of the American Physical Society

# APS April Meeting 2019

## Volume 64, Number 3

## Saturday–Tuesday, April 13–16, 2019; Denver, Colorado

### Session Z08: Particle & Fields Theory III

3:30 PM–5:06 PM,
Tuesday, April 16, 2019

Sheraton
Room: Governor's Square 10

Sponsoring
Unit:
DPF

Chair: Keping Xie, Southern Methodist University

### Abstract: Z08.00005 : Harmonic Triangles

4:18 PM–4:30 PM

View Presentation Abstract

#### Presenter:

Theodore M Lach

(Bell Labs Retired)

#### Author:

Theodore M Lach

(Bell Labs Retired)

^{th}2:3, and Perfect 4

^{th}3:4. I extended this concept to the triangle and gave it the name Harmonic Triangle. Consider a Rt Triangle ABC, with B being the Rt angle and AC the hypotenuse. A line from B to line AC, is the height of the Rt Triangle intersecting at D. Then by geometry the geometric mean of AD and DC is equal to the height BD for all angles Theta at C. Only certain values of Theta produce a critical angle that makes the triangle Harmonic. The first of these is Theta = 26.565 degrees, which happens when AD is ½ BD and BD is ½ DC. This is the first-generation Harmonic triangle. Similarly, you can construct the triangles for the perfect 5

^{th}and the perfect 4

^{th}. The perfect 5

^{th}has a critical angle of 33.6900 degrees. Now going back to the CBM, we observe that the cube root of a geometric series is also geometric. So, the cube root of the masses determined in that model are also related to each other by the geometric mean. One observes that these cube roots recreates the ratios 1:2, 2:3 and 3:4 which determine the critical angles. So, I contend that this supports the hypothesis made in the CBM and predicts that the masses of the quarks are a function of the radii

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