Sacred Geometry of the Starcut Diagram: The Genesis of Number, Proportion, and Cosmology
Autor Malcolm Stewarten Limba Engleză Hardback – 26 oct 2022
• Shows how the starcut diagram underlies the shaman’s dance in China, the Vedic Fire Altar in India, Raphael frescoes, labyrinth designs, the Great Pyramid in Egypt, and the building of ancient cities
• Explains how the starcut diagram was used in building and design, how it relates to Pythagoras’s Tetrakys, and how it contains knowledge of the Tree of Life
As Malcolm Stewart reveals in this lavishly illustrated study, the simplesquare figure of the Starcut diagram, created only with circles, has extraordinary geometric properties. It allows you to make mathematically exact measurements and build perfectly true level structures without a computer, calculator, slide rule, plumb bob, or laser level. Sharing his extensive research, along with hundreds of detailed diagrams and technical illustrations, the author shows how the Starcut diagram was the key to the building of humanity’s first cities and how it underlies many significant patterns and proportions around the world.
Using circles drawn from the vesica piscis, Stewart explains how to create the Starcut diagram and shows how this shape was at the foundation of ancient building and design, illustrating the numerous connections between the diagram and the creation of mandalas and yantras, stained glass windows, architectural ground plans, temples and other sacred buildings, and surveying methods. He also shows how the Starcut diagram reveals ancient geometric knowledge of pi, the Fibonacci sequence, Pythagorean shapes and seals, the golden ratio, the power of 108 and other sacred numbers, and magic squares.
Exploring the Starcut diagram’s cosmological and theological implications, Stewart explains how it contains knowledge of the Tree of Life and the Kabbalah. He examines how it relates to the Tetraktys, the key teaching device of Pythagoras, and other cosmograms. Demonstrating the ancient relationships existing between number, geometry, cosmology, and musical harmony, the author shows how the simple shape of the Starcut diagram unifies the many threads of sacred geometry into one beautiful mathematical tapestry.
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Specificații
ISBN-13: 9781644114308
ISBN-10: 1644114305
Pagini: 320
Ilustrații: Full-color throughout
Dimensiuni: 203 x 254 x 33 mm
Greutate: 1.04 kg
Ediția:2nd Edition, New Edition of Patterns of Eternity
Editura: Inner Traditions/Bear & Company
Colecția Inner Traditions
ISBN-10: 1644114305
Pagini: 320
Ilustrații: Full-color throughout
Dimensiuni: 203 x 254 x 33 mm
Greutate: 1.04 kg
Ediția:2nd Edition, New Edition of Patterns of Eternity
Editura: Inner Traditions/Bear & Company
Colecția Inner Traditions
Notă biografică
Malcolm Stewart is an author, musician, and designer who has created mosaic, stained glass, landscape, and interior designs for a range of secular and sacred environments. He is a former priest, BBC television producer, UN project director, and administrator for Ockenden Venture, a refugee relief charity. He has lectured widely throughout the UK, Europe, the US, and Canada. He lives in Tilford, Surrey, England.
Extras
Chapter 3: Opening the Box
“Some symbols always mean the same thing.”
--Alan Moore, Promethea
In architecture, city planning, furniture, pavements, game boards, wall tiles, plaid designs, graphs, grids, and weaves--the geometric square is everywhere. One speculates that knowledge of the square form goes all the way back to some ancient ancestor who first wove rushes or scored lines in the sand. Certainly by 20,0 BCE or so a Solutrean cave artist had inscribed square figures on the wall of the Cosquer Cave in southern France (see fig. 6.4, p. 64).
The square as a mathematical function is everywhere too, as the multiplication of a number by itself and, in square root form, as the unique factor that, multiplied by itself, produces a given number. Squaring is a feature of formulas for areas of circles and spheres; for geometric theorems; for coordinate systems; in the inverse square law; in laws concerning vacuums, gases, gravitation, radiation, acceleration, and more. Laws of mathematical logic and laws of nature are full of squares. Einstein’s famous E = mc2 is only the most famous equation that involves squaring. In that connection it was once commented to me that whereas some people are most interested in E (energy), others in m (solid matter), others in c (speed and light), I seem to have got stuck with the bit that most people hardly notice--the little number “2 ”. Squaring, however, is the “power” of a number, as defined by mathematicians. The ancient world, too, had it right: “qualitatively” the square denotes the manifest result that comes from the creative dynamism of the triad. Squaring adds magnitude to simple length. Whenever one sees the symbol √ or a superscript number (e.g., √31 or 1237) the squaring function is implied. A line of unit length becomes a plane of unit area when squared.
Figure 3.2 is a unit square, that is, the side length is 1. It has some immediate implications. Simply in the drawing of a square, another length, that of the diagonals, is suggested by the relationship of the opposite corners with each other. The diagonal of the unit square is a line that reveals its nature when it is itself squared; it becomes the side of a square twice the area of the original unit square. This new square has an area of 2, as is shown by counting up the component triangles. The line of its side is thus “the root of a square of value 2”--therefore “the square root of 2” (√2).
There is an immediate numerical mystery. The actual length of this diagonal produces the decimal 1.4142135 . . . , a number that goes on interminably and never falls into a repeating pattern. This means that the decimal part cannot be expressed as an exact fraction because it never settles down. A fraction is a ratio between the number above the line and that below. It is said to have been Pythagoras himself who first realized that √2 was neither a whole number nor a fractional ratio and was therefore “irrational,” meaning literally “without ratio.” It is reported also that Pythagoras only taught his private group (the mathematikoi) about irrationals and the related qualities of the pentagon--which, as we shall see, fairly seethes with another irrational: √5. Apparently one of the private group talked indiscreetly to some of the public group (the akousmatikoi or “hearers”). Versions of the story have Hippasus, the indiscreet student, being put to sea in an open boat, banished, or even murdered--which one doubts--by Pythagoras himself.
Though the Pythagoreans may not have known it, that diagonal and its awkward length was known more than a thousand years before in Mesopotamia, from where we even have a clay tablet that gives a solution of √2 correct to five decimal places!
Returning to basics, what actually is a square? Can it be dismantled beyond its obvious familiar and rather bland face? It certainly can, and this book is the outcome of just that.
Despite my grumbles about the way geometry was taught in school, we had a mathematics teacher, Mr. Ronald Hunt, who taught us something very important. At the top of his voice he would shout the word “WOMEK!” The word was an acronym from the initial letters of “write or mark everything known.” It is a key to finding fresh information. When we do this we are much more likely to pick out potential developments latent in the geometry. When, for instance, in order to get a perpendicular bisector, we draw a full vesica, rather than mean little arcs, we are “womeking.”
We can apply this to our main topic, which is simply an ordinary square--a flat box with four equal sides. Drawing such a square we may give each side some chosen length--1 being the simplest. Then having marked the right angle and put in the diagonals it might seem that we have finished. But in fact we know far more than this because a square is itself a construction. It has an internal rationale that unfolds in simple steps from the vesica that is the root/womb of all geometry. In its unfolding, it carries information with it. The sand reckoner’s square is generated by seven similar circles that establish nine points. One only notices this if one “womeks” the process by fully drawing everything in as it becomes relevant. It can be drawn very simply, by rotating a forked stick in the sand and stretching and snapping down a cord for the straight lines; such methods have been with us from very ancient times.
It is now as though we have lifted the surface of our square, like a hinged lid, to see its ancestral inner workings. In actually making it one has learned far more than is immediately obvious from just looking at it as an equal-sided box shape. The lattice triangulates the square in a way that is the key to all that follows. The triangle is the basic straight-line plane. Two points establish a line. Three points--unless all on the same line--establish a triangular plane. A square, of course, can be cut into triangles in an infinite number of ways; but here using the guide points from the square’s own generation, we get a natural, economic, and uniquely elegant triangulation that is truly loaded with information. There are two forms of the lattice (fig. 3.6 and 3.7); sometimes it is useful also to retain the central circle that was part of the construction (fig. 3.7).
It works as a mnemonic device for a vast swath of lore from the metric and sacred geometry of India, Mesopotamia, Egypt, China, Greece, Judea, Islamic culture, and Renaissance Europe. What is more, if one forgets its content, one has only to observe its measures, numbers, and proportions and everything reveals itself again.
I first came across it by chance as a possible grid underlying an Indian temple ground plan, of which I will tell you more in the next chapter.
“Some symbols always mean the same thing.”
--Alan Moore, Promethea
In architecture, city planning, furniture, pavements, game boards, wall tiles, plaid designs, graphs, grids, and weaves--the geometric square is everywhere. One speculates that knowledge of the square form goes all the way back to some ancient ancestor who first wove rushes or scored lines in the sand. Certainly by 20,0 BCE or so a Solutrean cave artist had inscribed square figures on the wall of the Cosquer Cave in southern France (see fig. 6.4, p. 64).
The square as a mathematical function is everywhere too, as the multiplication of a number by itself and, in square root form, as the unique factor that, multiplied by itself, produces a given number. Squaring is a feature of formulas for areas of circles and spheres; for geometric theorems; for coordinate systems; in the inverse square law; in laws concerning vacuums, gases, gravitation, radiation, acceleration, and more. Laws of mathematical logic and laws of nature are full of squares. Einstein’s famous E = mc2 is only the most famous equation that involves squaring. In that connection it was once commented to me that whereas some people are most interested in E (energy), others in m (solid matter), others in c (speed and light), I seem to have got stuck with the bit that most people hardly notice--the little number “2 ”. Squaring, however, is the “power” of a number, as defined by mathematicians. The ancient world, too, had it right: “qualitatively” the square denotes the manifest result that comes from the creative dynamism of the triad. Squaring adds magnitude to simple length. Whenever one sees the symbol √ or a superscript number (e.g., √31 or 1237) the squaring function is implied. A line of unit length becomes a plane of unit area when squared.
Figure 3.2 is a unit square, that is, the side length is 1. It has some immediate implications. Simply in the drawing of a square, another length, that of the diagonals, is suggested by the relationship of the opposite corners with each other. The diagonal of the unit square is a line that reveals its nature when it is itself squared; it becomes the side of a square twice the area of the original unit square. This new square has an area of 2, as is shown by counting up the component triangles. The line of its side is thus “the root of a square of value 2”--therefore “the square root of 2” (√2).
There is an immediate numerical mystery. The actual length of this diagonal produces the decimal 1.4142135 . . . , a number that goes on interminably and never falls into a repeating pattern. This means that the decimal part cannot be expressed as an exact fraction because it never settles down. A fraction is a ratio between the number above the line and that below. It is said to have been Pythagoras himself who first realized that √2 was neither a whole number nor a fractional ratio and was therefore “irrational,” meaning literally “without ratio.” It is reported also that Pythagoras only taught his private group (the mathematikoi) about irrationals and the related qualities of the pentagon--which, as we shall see, fairly seethes with another irrational: √5. Apparently one of the private group talked indiscreetly to some of the public group (the akousmatikoi or “hearers”). Versions of the story have Hippasus, the indiscreet student, being put to sea in an open boat, banished, or even murdered--which one doubts--by Pythagoras himself.
Though the Pythagoreans may not have known it, that diagonal and its awkward length was known more than a thousand years before in Mesopotamia, from where we even have a clay tablet that gives a solution of √2 correct to five decimal places!
Returning to basics, what actually is a square? Can it be dismantled beyond its obvious familiar and rather bland face? It certainly can, and this book is the outcome of just that.
Despite my grumbles about the way geometry was taught in school, we had a mathematics teacher, Mr. Ronald Hunt, who taught us something very important. At the top of his voice he would shout the word “WOMEK!” The word was an acronym from the initial letters of “write or mark everything known.” It is a key to finding fresh information. When we do this we are much more likely to pick out potential developments latent in the geometry. When, for instance, in order to get a perpendicular bisector, we draw a full vesica, rather than mean little arcs, we are “womeking.”
We can apply this to our main topic, which is simply an ordinary square--a flat box with four equal sides. Drawing such a square we may give each side some chosen length--1 being the simplest. Then having marked the right angle and put in the diagonals it might seem that we have finished. But in fact we know far more than this because a square is itself a construction. It has an internal rationale that unfolds in simple steps from the vesica that is the root/womb of all geometry. In its unfolding, it carries information with it. The sand reckoner’s square is generated by seven similar circles that establish nine points. One only notices this if one “womeks” the process by fully drawing everything in as it becomes relevant. It can be drawn very simply, by rotating a forked stick in the sand and stretching and snapping down a cord for the straight lines; such methods have been with us from very ancient times.
It is now as though we have lifted the surface of our square, like a hinged lid, to see its ancestral inner workings. In actually making it one has learned far more than is immediately obvious from just looking at it as an equal-sided box shape. The lattice triangulates the square in a way that is the key to all that follows. The triangle is the basic straight-line plane. Two points establish a line. Three points--unless all on the same line--establish a triangular plane. A square, of course, can be cut into triangles in an infinite number of ways; but here using the guide points from the square’s own generation, we get a natural, economic, and uniquely elegant triangulation that is truly loaded with information. There are two forms of the lattice (fig. 3.6 and 3.7); sometimes it is useful also to retain the central circle that was part of the construction (fig. 3.7).
It works as a mnemonic device for a vast swath of lore from the metric and sacred geometry of India, Mesopotamia, Egypt, China, Greece, Judea, Islamic culture, and Renaissance Europe. What is more, if one forgets its content, one has only to observe its measures, numbers, and proportions and everything reveals itself again.
I first came across it by chance as a possible grid underlying an Indian temple ground plan, of which I will tell you more in the next chapter.
Cuprins
Acknowledgments
Prologue
1 An Introduction to Sacred Geometry: Who Tastes Knows
2 Unity in the Liberal Arts
3 Opening the Box
4 Altar of the First Fire
5 The Tartans of Pythagoras
6 Pyramid Connections
7 The Sword Master’s Floor Plan
8 The Hidden Geometry of the “Divine Raphael”
9 Guardian Apollo
10 Conjecture
11 Beautiful Diagonals
12 Clues to a Forgotten Lore
13 The Mother of All Pentagons
14 The Area Key
15 The Secret and the Sacred
16 Wheels of Number and Time
17 Powers of 108
18 Ten Pebbles in the Sand
19 Harmony at the Horizon of Being
20 The Shape of Concord
21 The Lyre of Apollo
22 The Puzzle That Drives You Mad
23 Reflections within the Light
24 Inevitable Sufis
25 The Luo Shu and the Limping Dance
26 . . . Finally
Epilogue
appendix
The Starcut Glass Bead Games
Bibliography
Index
Prologue
1 An Introduction to Sacred Geometry: Who Tastes Knows
2 Unity in the Liberal Arts
3 Opening the Box
4 Altar of the First Fire
5 The Tartans of Pythagoras
6 Pyramid Connections
7 The Sword Master’s Floor Plan
8 The Hidden Geometry of the “Divine Raphael”
9 Guardian Apollo
10 Conjecture
11 Beautiful Diagonals
12 Clues to a Forgotten Lore
13 The Mother of All Pentagons
14 The Area Key
15 The Secret and the Sacred
16 Wheels of Number and Time
17 Powers of 108
18 Ten Pebbles in the Sand
19 Harmony at the Horizon of Being
20 The Shape of Concord
21 The Lyre of Apollo
22 The Puzzle That Drives You Mad
23 Reflections within the Light
24 Inevitable Sufis
25 The Luo Shu and the Limping Dance
26 . . . Finally
Epilogue
appendix
The Starcut Glass Bead Games
Bibliography
Index
Recenzii
“Malcolm Stewart has written an original and finely produced book about the structural invariance of number, seen most clearly as sacred geometry but especially clear in his Starcut diagram.”
"An absolute 'must' for students and practitioners of Geometry, Toplogy, Cosmology, Sacred Geometry of the Starcut Diagram: The Genesis of Number, Proportion, and Cosmology is beautifully and profusely illustrated in full color throughout. This coffee-table style (8 x 1.3 x 10 inches, 1.69 pounds) volume is certain to be a treasured and unique addition to personal, professional, community, and academic library Sacred Geometry and Metaphysical Studies collections. It should be noted for students, academia, and non-specialist general readers with an interest in the subject that Sacred Geometry of the Starcut Diagram is also available in a digital book format (Kindle, $26.99)."
“I remain in awe of this book. Sacred Geometry and the Starcut Diagram is probably the single most important addition to the body of sacred geometry to re-emerge in a decade. Malcolm Stewart’s exposé of this deceptively simple device will have scholars of art, architecture, mathematics, and geometry entranced for centuries to come.”
"Filled with beautiful artwork, pictures, and illustrations, the reader is guided step-by-step in the workable knowledge of the ancient Starcut diagram. Furthermore, Stewart assures the readers/students that they will be able to make precise mathematical measurements and build perfect structures without a computer, calculator, slide rule, plumb bob, or a laser level."
"An absolute 'must' for students and practitioners of Geometry, Toplogy, Cosmology, Sacred Geometry of the Starcut Diagram: The Genesis of Number, Proportion, and Cosmology is beautifully and profusely illustrated in full color throughout. This coffee-table style (8 x 1.3 x 10 inches, 1.69 pounds) volume is certain to be a treasured and unique addition to personal, professional, community, and academic library Sacred Geometry and Metaphysical Studies collections. It should be noted for students, academia, and non-specialist general readers with an interest in the subject that Sacred Geometry of the Starcut Diagram is also available in a digital book format (Kindle, $26.99)."
“I remain in awe of this book. Sacred Geometry and the Starcut Diagram is probably the single most important addition to the body of sacred geometry to re-emerge in a decade. Malcolm Stewart’s exposé of this deceptively simple device will have scholars of art, architecture, mathematics, and geometry entranced for centuries to come.”
"Filled with beautiful artwork, pictures, and illustrations, the reader is guided step-by-step in the workable knowledge of the ancient Starcut diagram. Furthermore, Stewart assures the readers/students that they will be able to make precise mathematical measurements and build perfect structures without a computer, calculator, slide rule, plumb bob, or a laser level."
Descriere
Stewart shows how the starcut diagram was the key to the building of humanity’s first cities and how it underlies many significant patterns and proportions around the world.