This is another brief expert from my book “Talking to the Birds”, I hope you enjoy it. I have included some notes from Ben Iverson as well.

Only a trained mathematician/geometer can recognize the missing elements in the geometric canon, that which has been ignored for the last 60 years, you cannot accidentally or casually find this most unique arithmetic, it is not based just on porisms. It is not number theory or invented mathematics, it does not need calculus, there is no approximations, it does begin its understanding as the trivial terms that most assume to be understood. Let us begin with the trivial terms so we can dispense with all the mystery.

Quantum Arithmetic uses a natural number system which has a base in of all of the prime numbers which occur in the problem you need to solve. All its numbers are interlocked in a geometric arrangement. All numbers are positive integers, no decimal points or irrational numbers. Addition, subtraction, multiplication and division is all that is mainly required, the use of square roots and diadic fractions are also used, they are mainly used to verify the results and find the whole number ratio’s. The ordering into Par numbers can also verify and help understand the internal arrangements.

There are sixteen primary identities. The first four are given the identity of “a”, “b” “e” and “d”. These, are the roots of its given problem, and is the base numbers. The next twelve identities are the upper case letters A through L. They are combinations, of the first four base numbers, and are usually considered as one dimension, higher than the base identities. They will denote linear dimension, surface areas, or volumes, for the three standard dimensions above the roots.

There is also one dimension below the “roots” b, e, d, and a. They are called “quaternions”, and are the square roots of the root numbers. In conventional mathematics, these are called “Gaussian Integers”. They are integers, only when their base number is a perfect square. A problem in QA is well defined and problems solvable when only one of the upper case identities is assigned a value and the name of that identity. All of the values within a problem in QA are intertwined this gives you hundreds of ways to solve any given problem.

The Quantum Number for one figure is the same for all geometric figures. A single quantum number defines the magnitude of the measurements, as they are related between themselves. Different geometric figures can be connected and their dimensions are calculated once for all of the various shapes. The shapes used are:

1 Right triangles

2 Equilateral triangle

3 Isosceles Triangles

4 Triplet circles (Koenig Series)

5 Triplet Squares (Koenig Series)

6 Ellipses

Each shape is calculated from a single Quantum Number. The one requirement is that the second and third integers of the Quantum Number must be prime to each other. These are in the Fibonacci sequence. They derive from Euclid VII, Proposition 28. The first and fourth integers become, “Sum and Difference” numbers.

All Quantum ellipses will give the value one when you apply the Steiner Inellispe porism. The numeric laws it contain can be geometrically reconstructed with s0me of the most excellent programs such as GeoGebra, First you need to understand the construction of the “Quantum Ellipse”. The usual 2 foci and semi-major length/third point will not suffice only Conic section ellipse will create the correct elliptical form, if you have 5 points that are aligned to the quantum arithmetic standard. I hope some of these diagrams will help clarify all the parameters. The best way to construct an ellipse of this sort is to start with a string with length 3+4+5=12 a Pythagorean triangle. This method was re-discovered by James Clerk Maxwell in the mid 1800’s.

This type of geometry had been developed by Ben Iverson from the 1950’s through to to 1990’s extended by Dale Pond and myself over the last 20 years.

After many calculations and with the application of Quantum Arithmetic, I have created a beginning for the foundation of the Quantum 120degree Isoceles. The smallest integer units begin at the 4 digit root numbers, which means some calculations are up to 20 numbers long This study has revealed the Quantum Equilateral also within its perimeter, having now multiple Z line integers.

Listed below is the some of the raw data for one of the smallest Quantum Isosceles 120 degree Triangle, the added Var1,Var2,Var3,Var4,Var5,Var6 are part of an incomplete data list, you can ignore them or just make each equal zero.

b = 2646 = d – e

e = 4343 = a – d

d = 6989 = b + e

a = 11332 = d + a

B = 7001316 = b x b

E = 18861649 = e x e

D = 48846121 = d x d

A = 128414224 = a x a

C = 60706454 = 2 x d x e

Fe = 29984472 = d x d – e x e + Var1

G = 67707770 = d x d + e x e

L = 151687580848524 = ( d x d x d x e – e x e x e x d ) / 6

H = 90690926 = ( a x a + 2 x b x a – a x a ) / 2

I = 30721982 = ( e x e + 2 x d x e ) – d x d

J = 18492894 = d x d – d x e

K = 79199348 = d x d + d x e

We = 109552575 = d x e + d x a

X = 30353227 = d x e

Ye = 79568103 = 2 x d x e + d x d + Var2

Ze = 98060997 = e x e + d x d + d x e + Var3

Wib = 328657725 = 3 x ( d x e + d x a )

Fi = 109552575 = d x e + d x a + Var4

Yi = 219105150 = 2 x ( d x e + d x a ) + Var5

Zi = 109552575 = d x e + d x a + Var6

Wis = 189750626 = ( d x e + d x a ) x sq 3

Wh = 94875313 = ( ( d x e + d x a ) / 2 ) x sq 3

Thank you

Regards Arto