# “The Rest of Euclid” (table of contents & foreword)

This page introduces the work of physicist Robert (Bob) L. Powell, Sr in the field of Mathematics.

This precedes the publishing of the web site http://TheRestOfEuclid.com and the forthcoming book:

”The REST of Euclid”

Robert L. Powell, Sr. drawing: “Seven x Three x Six, Sat. 6/25/88

**Photo Courtesy: Gregory B. Gerran,**

**©**1990**ROBERT LEE POWELL, SR.**

*Born: Kerns, Texas*

*Education: Fisk University, Nashville, TN Degree: B.S. & M.S., Physics*

His knowledge and expertise also extends to some of the traditions of African art in the application of mathematical ratios and principles of Euclidian Geometry. His knowledge and lectures on the subject have influenced and changed the lives of many artists including the late Dr. John Thomas Biggers, and aspiring artists in colleges and universities in the U.S. and around the world. In his 1997 visual mathematics installation “In This House,” a part of the **Project Row Houses** in Houston Texas, he delivered a multi-level system of ancient mathematical teachings from a diverse mixture of sub-cultures. His cutting-edge mathematics uses the rules and tools of Euclid, and the geometrical roots of a quadrivium: Sacred art, Sacred architecture, Modernity’s physics, and Modernity’s biology. Powell, co-author of the yet-to-be-published “The REST of Euclid”, and now octogenarian resides in Greensboro, NC and Houston, TX.

*email: DocPowell@juno.com and TheRestOfEuclid@gmail.com*

*Land Mail: 1507 Hamblen Street, Houston TX 77009*

*and 2128 Wright Ave, Greensboro, NC 27403-1635*

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The following text is excerpted from a paper written by Robert L. Powell, Sr:

**Toward the Content Design for a T-STEM Center** Thursday 3/9/2006 R. L. P., Sr.

**The Serendipitous Discovery of the Fractal Integer Arithmetic.**

In mid-1978 a professional artist colleague, Professor John Biggers of the Texas Southern University School of Art presented Powell with an urgent and insistent aesthetic inquiry in connection with the *qualitative* embodiment of the quantitative relationship that Leonardo da Vinci had referred to as the *Divine* Proportion. Biggers desired to discover some of the special *gestalt* which this relation endowed in the canonical works of ancient Artists and Architects, as admired by da Vinci.

The inquiry led to a several week duration study, in Biggers’ Art School, of the geometry rules which seemed to *govern* and to *guide *the composers of the ancient compositions in Art and Architecture. The several week long study enabled Biggers to establish a new and *profound *personal and professional connection with the canon of geometry rules which seemed to guide and govern the ancient composers’ works of Art and Architecture. The code so admired by da Vinci was recognized to be merely *the most elegant* of a hierarchy of *only three* codes which governed and guided the holistic integrity of *all* the canonical works of ancient *Sacred Geometry*^{ (1) }

A meta-mathematical study and synthesis of the three codes, by Powell, led to the recognition that the three codes formed the hierarchy of expression of *an ancient Theorem of Euclid*, a Theorem of Euclidean Geometry which *calibrates* the entire Euclidean *plane* , in sets of Cartesian coordinates; the hierarchy of coordinates calibrates the plane, with positional notation, in terms of (±) powers of the three *fractal number integer ***vectors** : √3 ; √2 ; and ½ [√5 + √1] .

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This table of contents and foreword are from a 2004 draft (a work-in-progress).

The intent of the following text is to introduce the forthcoming book:

**THE REST ** **of EUCLID:**

**An Ancient Architecture of Arithmetic and the Modern Theory of Number**

[*draft of April 30, 2004]*

by

Robert L. Powell, Sr., Robert L. Powell, Jr., and Vandorn Hinnant III

The G.R. Lomanitz Laboratory of *Visual* Mathematics

The Practical Science Institute

1507 Hamblen Street, Houston Texas 77009

and

409 South Chapman Street, Greensboro North Carolina 27403, USA

e-mail: docpowell@juno.com

©2004 by Robert L. Powell, Sr.

*All rights reserved. This book may not be reproduced in whole or in part, or transmitted in any form, without the written permission from the above-identified copyright holder, except by a reviewer who may quote **brief passages in a review; nor may any part of this book be stored in a retrieval system, or **transmitted in any form or by any means electronic, mechanical, photographic, **photocopy, or by any other method without the written permission **of the above-identified copyright holder.*

TABLE OF CONTENTS

THE *REST* OF EUCLID

FOREWORD p.2

INTRODUCTION p.5

**Part I. THE REST OF ‘EUCLID’**

**Chapter I. DEFINITION OF THE Non-ELEMENTARY NUMBER THEORY **p.7

The circle number integers.

A countable set of *circumference numbers*;

Its corollary, a countable set of circumscribed *area numbers*;

Their corollary partner, a countable set of *radius quardrature numbers*.

**Chapter II. FRACTAL INTEGER NUMBER ANALYTIC GEOMETRY **p. 10

Countable partitions of the circle number *integers* ( the transcendental number integers),

( i.e., Gauss’ Cyclotomy )

**Chapter III. A PLANAR QUALITATIVE PROPERTY OF NUMBER. **p. 12

The Planar Addition Law for the quartet of radii sets ( orthogonality of the *vector*

radius quartet sets, and the *corollary* planar addition rule ).

**Chapter IV. DERIVATION OF A FRACTAL INTERGER NUMBER ****TRIGONOMETRY** **TABLE** p. 14

** **Determined by the symmetry properties of the eigen-set of radii.

**Chapter V. DERIVATION OF THE THREE PLANE SPACE-TILING FRACTAL INTEGER NUMBER ORTHOGONAL COORDINATE SYSTEMS ** p. 19

The hierarchy of Fractal Number Integer coordinate systems for the Euclidean plane.

Gauss’ Cyclotomy for the ‘*trans*-rational’ radii numbers.

- The powers of [ Ö3]
^{± 1}, vs [Ö1] ; - The powers of [Ö2]
^{± 1}, vs [Ö1] ; - The powers of ½ [ Ö5 ± Ö1] , vs [ Ö1]

**Chapter VI. THE HINNANT-POWELL FRACTAL NUMBER SERIES** p.23

Some Fractal Number Interpolations, generalizations of the Fibonacci series and

the Lucas series, as Scaled Calibrations of the Euclidean plane.

**Part II. SOME IMPLICATIONS OF Part I ( i.e., THE REST OF ‘EUCLID’ )**

**FOR 20 ^{TH} CENTURY MATHEMATICS**

### Chapter VII. The Circumference Number Integers, The Fractal Number Integers and **THEIR IMPLICATIONS FOR 20**^{TH} CENTURY MATHEMATICS, *qua MATHEMATICS ** ***p.32**

^{TH}CENTURY MATHEMATICS,

*qua MATHEMATICS*

Implications for the Number *line* calibrations; – for arithmetic; – for algebra;

– for analytic geometry; – for kinematics; – for analysis/synthesis

– for Sommerfeld’s Spatial Frequency Analysis.

**Chapter VIII. The Transcendental Number Integers, and ****THEIR IMPLICATIONS FOR 21 ^{ST} CENTURY MATHEMATICS, **

**p.39**

*qua**APPLIED**MATHEMATICS*COMPUTER SCIENCE

Hardware;

Software;

*Wetware*

INCIPIENT ‘*NEW’ *DIGITAL NUMBER COMPUTATION STRATEGIES.

The Renormalization Group method;

Wavelet analysis-synthesis;

Cyclostationarity

### Part III. SOME IMPLICATIONS OF Part II **FOR 21**^{ST} CENTURY SCIENCE

^{ST}CENTURY SCIENCE

**Chapter IX. BIOLOGY OF THE LIVING CELL CHIRALITY **p. 45

The fourth state of matter: the Theory of the Number *Plane* is necessary and sufficient for

theoretical models and experimental design parameters for this complexity.

**Chapter X. NANOTECHNOLOGY and BIOMIMICRY **p. ?

**Chapter XI. PHYSICS **p. ??

**I**nherently non-linear classical kinematics & dynamic precise mathematical representations for String Theory

**Part IV. SOME IMPLICATIONS OF ‘ THE REST OF ‘EUCLID’**

**as PARADIGM FOR a THIRD MILLENNIUM POLITICS OF ECOLOGY**

**Chapter XII. **p. ??

** **John D. Rockefeller 3

^{rd }Albert Szent Gyorgyi

Jonas Salk Riane Eisler

Jeremy Rifkin Norbert Wiener

euclid 6 -2-

# FOREWORD

1. WE ARE a physicist, an architect, and a *visual* artist – a trans-disciplinary team of non-mathematician *intellectuals*. Our connection is a complementary set of professional interests in the study of a *trans-intellectual* question: by what methods were the ancient guilds of architects and artists so able to embody into their works the *gestalt* element of beauty and fitness that characterizes their agelessly elegant compositions?

A quite adequate capture of the ‘flavor’ of the elusive aesthetic property we study can be inferred from our paraphrase of a set of remarks by Professor Dan Pedoe – himself a careful student of the guilds. Early in his *pleasure*– encouraging book, *GEOMETRY and the VISUAL ARTS*, Pedoe gives a review of : *the appropriate breadth and depth of concern with the whole subject of architecture*, as recommended by Marcus Vitruvius – himself one of the significant links in our unbroken chain of inherited guildsmen.^{(1)} Our paraphrase:

The elegant construction depends on *order, arrangement, eurhythmy, propriety, symmetry, and **economy*. …. Order gives due measure to the members of a work considered separately. Symmetry gives agreement to the proportions of the whole. It is an adjustment in according to quantity. By this is meant the selection of modules from the members of the work itself, and constructing the whole work to correspond. Eurhythmy is beauty and fitness in the adjustment of the members of a work.

[… *and*… ]

*Symmetry is* a proper agreement between the members of the work itself, and the relation between the different parts of the whole general scheme,

*in accordance with a certain part*

**selected**as*.*

**the standard**Any search for the technical methods by which these ineffable aesthetic embodiments into *visual structures* are accomplished *must* lead an interest such as ours to a study of the guilds’ *strategic* employment of their three essential instruments for producing geometric constructions in *the plane*:

the compass;

the straight edge;

*and *[!!!]

an *un-marked *plane surface, of some sort.

Our experimental studies of the ancients’ works, using the *three* essential tools, compelled us to consider, as a working proposition, that the guildsmen were governed rigorously by a protocol of rules which nevertheless permitted, perhaps indeed *guaranteed* the consequent embodiment of the ineffable *qualities* captured in the Vitruvius precis. His remark, “ …. in accordance with a certain part *selected* as *the standard*”, was particularly compelling as an organizing clue to guide our experimental strategies at de-constructing ^{(2)}, and re-constructing, the Implicate Order plane geometry patterns which seemed to govern the general scheme of the evolution of a work.

Indeed, this snippet from Vitruvius was found to explain *precisely* the key to the success of our program of gestalt pattern analysis and synthesis: in over a couple of decades of playful study we slowly learned that the ancients merely *required* that their works be *Theorems* of *plane* , and 3-dimensional, *Geometry* !!

euclid 6 -3-

That is to say, the Implicate Order *pattern* by which the guilds assured the gestalt coherence of the whole general scheme was an Implicate Order *palimpsest* determined strictly by the Five ‘Rules of Euclid’ which govern and assure the construction of a Theorem. We recognized that this system of rules is the *complete* set of Canons that *governed * and *guided* their choices of composition strategies and tactics for relating between the different parts of the whole general scheme, *in accordance with a certain part selected* *as a reference*. We recognized that centuries of canonical constructions – say, from ancientEgypt’sTemple ofKarnak toNew York City’s Cathedral of Saint John the Devine – give enduring record of the guilds’ adherence to the perennial accumulation of a syllabus of strategic, canonical, gestalt-embodying palimpsests, governed by this protocol.

2. This SET OF (FIVE) RULES comprises three of the five ‘Postulates of Euclid’, which together with the five “Axioms of Euclid’, provide the platform for Euclid’s masterpiece, *The Elements*. One of those unique books like the Bible which seem to fuse the best efforts of generations of creative minds into a single inspired, creative whole, *The Elements* is a work of such commanding lucidity and style that some scholars consder it the most coherent collection of closely reasoned thoughts ever set down by man ^{(3)}

*The Elements* contains 13 books, or chapters, which describe and prove a good part of all that the human race knows, even now, about lines, points, circles and the elementary three-dimensional shapes. ^{(3)}

3. Our program of analysis and synthesis of the ancient palimpsest structures has permitted the recognition of a related pair of Theorems which is the foundation for the construction of *all possible* Implicate Order patterns embodying the gestalt eurhythmy, symmetry, and economy. This inter-articulated pair of theorems extends, qualitatively and quantitatively, what the human race knows about lines, points, circles and the elementary three-dimensional shapes. Our extension of this knowledge, our ancient pair of theorems, emerges as implications embodied in the teachings of *very first Problem* of *the very first Book* of

*The Elements*.

To our great surprise, the pair of theorems *also* provides – permits the definition of- a remarkable *generalization *of the foundation of the architecture of the structure of our modern 20^{th} century mathematics.

The revolutionary generalization also, of course, *evolutioinizes *applied mathematics as a 3^{rd} millennium scientific instrument.

We refer to this ancient pair of implications of Proposition 1, Book I – ignored, neglected, over looked for two millennia now – as THE REST OF EUCLID.

4. The purpose of our book, then, is two-fold: (1) to present this serendipitous discovery of a *revolutionary* generalization of the elementary mathematical concept, Number, and its co-related *evolutionary* mathematical architecture, an *ab inito* non-linear Arithmetic; and (2) a cursory sketch of the *evolutionary *applied mathematics instrumentations co-related to the richer Arithmetic.

5. For cultural reasons, the professional mathematician must, as a conditioned reflex, play Devil’s Advocate to the claim of discovery, by a trio of non-mathematician amateurs, of such a humongous un-tapped epistemological universe for the Western Europe intellect as (1) our THE *REST* OF EUCLID and (2) its co-related evolutionary applied mathematics enablements. Surely the ‘West’-dominated guild of mathematicians has already *fully* exhausted the Teachings of *The Elements* as a rational source of what is possible to know.

euclid 6 -4-

Therefore, so as not to be dismissed peremptorily by the likes of Ian Stewart and John Casti as a ‘Math Crank’ ^{(4)}, our book’s presentation has tried to adhere to that profession’s arcane ( and in key places, now obsolescent) vocabulary, grammar and rhetoric. Nevertheless we intend the interested non-mathematician reader to be able to grok the simple revolutionary and evolutionary features of the generalizations. Hence the presentation seeks also to subvert the conditioned reflex aversion-to-mathematics of the non-mathematician reader.

…………………..

(1) Pedoe, Dan. *GEOMETRY and the VISUAL ARTS*.New York;Dover 1983. p. 18.

(2). Derrida, J. *Edmund Husserl’s Origin of Geometry: An Introduction*. Stoney Brook: Nicholas Hays, Ltd. 1978.

(3). David Bergamini, et al [Rene Dubos, Henry Margenau, C.P. Snow], eds. *Life Science Library MATHEMATICS*.New York: TIME INCORPORATED. 1963. p. 45, ff.

(4). George Johnson. “Genius or Gibberish? The Strange World of the Math Crank”, New York Times,

Tues, Feb 9, 1999, p. D-1.