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

“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”

"Seven x Three x Six" (#2)

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

Doc Powell in study, Houston TX, 1990

Photo Courtesy: Gregory B. Gerran, © 1990
Born: Kerns, Texas
Education: Fisk University, Nashville, TN Degree: B.S. & M.S., Physics
Bob Powell is a clandestine “national treasure” in the world of mathematics, and physics. A veteran teacher at Fisk University, Nashville, TN; Lowell Technical Institute, Lowell, MA; Oakland University, Rochester, MI, and Texas Southern University, Houston, TX has worked for a variety of corporations. This phenomenal physics professor is the co-inventor of the esoteric field of work, Holographic Non-Destructive Testing, which is arguably the most beautiful and appropriate engineering and technological use of the hologram-making process so far discovered.

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


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] .


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:


An Ancient Architecture of Arithmetic and the Modern Theory of Number

[draft of April 30, 2004]


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


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.



FOREWORD                                                                                                                                                 p.2

INTRODUCTION                                                                                                                                         p.5



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.


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

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

  1. The powers of [ Ö3]± 1 ,  vs  [Ö1] ;
  2. The powers of [Ö2]± 1 , vs  [Ö1] ;
  3. 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.





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

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 21ST CENTURY MATHEMATICS, qua APPLIED MATHEMATICS                       p.39






The Renormalization Group method;

Wavelet analysis-synthesis;



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. ??

Inherently non-linear classical kinematics & dynamic precise mathematical representations for String Theory





Chapter XII.                                                                                                                                           p. ??

                        John D. Rockefeller 3rd           Albert Szent Gyorgyi

Jonas Salk                               Riane Eisler

Jeremy Rifkin                          Norbert Wiener


euclid 6                                                                                                                                                           -2-


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 20th century mathematics.

The revolutionary generalization also, of course, evolutioinizes applied mathematics as a 3rd 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.