ARC Corpus · Volume 2
The Figure
Two circles and a square: geometry, the forgotten operations and the convergences of the half
« Le cercle conserve ; le carré inverse et forme. »
Filipe Bastos, Le Carré-Illusion
Abstract
Everything the ARC corpus says about its root figure: two concentric circles with a square between them. The area of the small circle is exactly half that of the large one, the diagonal of the square is the diameter of the large circle, and the three constants √2, π and φ find their place in it. The volume covers the seven regular polygons from triangle to nonagon and the geometric origin of units of measure, the historical confusion of double and square, the other triangle, the notified square, the operations the author calls forgotten, the beat, trinitarian subtraction, the octet rule, the end of integral calculus and the seven convergences of one half, and the fractal law. It also establishes the uniqueness of base ten through the square closure 9 = 3², and studies the family of bases that only share its form.
Keywords: geometry, two circles and a square, regular polygons, square root of two, pi, golden ratio, octet rule, base ten
The chapters themselves are in French. Open any chapter and use the Translate links in the header for a machine translation; the abstract and the table of statements on this page are translated by hand.
Résumé (français)
Tout ce que le corpus ARC dit de sa figure racine : deux cercles concentriques et un carré coincé entre eux. L'aire du petit cercle y est exactement la moitié de celle du grand, la diagonale du carré est le diamètre du grand cercle, et les trois constantes √2, π et φ y trouvent leur place. Le volume déroule les sept polygones réguliers du triangle au nonagone et l'origine géométrique des unités de mesure, la confusion historique du double et du carré, l'autre triangle, le carré notifié, les opérations que l'auteur dit oubliées, le battement, la soustraction trinitaire, la règle de l'octet, la fin du calcul intégral et les sept convergences du demi, et la loi fractale. Il établit aussi l'unicité de la base 10 par la clôture carrée 9 = 3², et étudie la famille des bases qui en ont seulement la forme.
Keywords: géométrie, deux cercles et un carré, polygones réguliers, √2, π, nombre d'or, règle de l'octet, base 10
Contents
- Foreword
- Chapter 1. The fundamental figure
- 1.1 Before the figure: the how is legible, the why is badly posed
- 1.2 The construction in four gestures
- 1.3 The square is derived: the projective illusion
- 1.4 The identity reading
- 1.5 The eight points, the octagon and the pentagram
- 1.6 The Platonic solids and the octahedron
- 1.7 The geometric trinity √2, π, φ
- 1.8 Hippasus, Stevin, Gauss: the three illusions
- 1.9 The four operators that come out of the figure
- Chapter 2. The three constants: √2, π, φ
- 2.1 The three Gates: √2, √3, √5
- 2.2 √2, operator of square-circle articulation
- 2.3 π: the primacy of the circle
- 2.4 Closed π: 22/7 = 11/3.5
- 2.5 φ: the pentagon and the golden number
- 2.6 Synthesis: three operators of one figure
- Chapter 3. The cadastre comes out of the figure
- 3.1 The nine digits are not a base
- 3.2 The multiplication table reduced to digital root
- 3.3 Why 9 ≠ 0: the loss of rank
- 3.4 Pascal's cascade
- 3.5 The two grids
- 3.6 The 11 designates, the 12 operates
- Chapter 4. Two circles and a square: the double and the square
- 4.1 The root: double and square, equal only at 2
- 4.2 The buried translator: √2
- 4.3 The minimal grammar: two bricks
- 4.4 The fermion/boson partition by ×3
- 4.5 The squares and the two operators
- 4.6 The sealed string and the quantum
- 4.7 The resurrection of √2: 2√2
- 4.8 The lineage of amputations as a single slip
- 4.9 Conclusion of the master document
- Chapter 5. The square-illusion and the geometric origin of units
- 5.1 Drawing by hand and constructibility
- 5.2 The universal law: every polygon carries the memory of 9
- 5.3 The triangle (n = 3): the geometry of time
- 5.4 The square (n = 4): the geometry of space and angle
- 5.5 The pentagon (n = 5): the geometry of life
- 5.6 The hexagon (n = 6): the geometry of matter
- 5.7 The heptagon (n = 7): the geometry of light
- 5.8 The octagon (n = 8): the geometry of energy
- 5.9 G, the 3 under its seven faces
- 5.10 The nonagon (n = 9): the geometry of closure
- 5.11 The complete system
- 5.12 The greatest error: measuring the fundamental instead of reading it
- 5.13 The pyramid as a folding of the figure
- 5.14 Conclusion of the Square-Illusion
- Chapter 6. The other triangle
- 6.1 Three anchors
- 6.2 The triangle that was never tried
- 6.3 Presence and status: the central distinction
- 6.4 k_arc = 1/2, cadastral hypotenuse
- 6.5 The crisis of incommensurables as a scale artefact
- 6.6 The sinusoidal paradox
- 6.7 Consequences
- 6.8 Three external anchors
- 6.9 The immense simplicity
- Chapter 7. The notified square
- 7.1 The thesis: the square seals the connections
- 7.2 The fibre theorem: what the square loses
- 7.3 Born's rule reread
- 7.4 The idempotence of the cost: the price is paid once
- 7.5 The content of the seal: the knot of the flanks and the walls
- 7.6 The rule: notification to the square
- 7.7 Open site
- Chapter 8. The forgotten operations and the beat
- 8.1 The square has four sides, we kept only one
- 8.2 The three killed sides and their scars
- 8.3 The beat: two operations, a single gesture
- 8.4 Where 4.5 comes from: the overdetermined point
- 8.5 The resolution of the irrational is a liberation of a third
- 8.6 The square, a suspended moment
- 8.7 Lavoisier corrected
- 8.8 The beat of the heart
- 8.9 Conclusion of the Beat
- Chapter 9. The trinitarian subtraction
- 9.1 Two subtractions
- 9.2 The 2 is a splitter: 1 − 1 does not make 0
- 9.3 The physical proofs, reread as readings
- 9.4 The dead square and the 4 → 8 → 5 torsion
- 9.5 Photosynthesis, subtraction of light
- 9.6 137 and 163: light and space
- 9.7 The subtractive circuit and the two descents
- 9.8 The gradient 198, the fall 981 and the rise 189
- 9.9 Strings and 496
- 9.10 The founding subtraction: 6 − 4 = 2
- 9.11 The universe subtracts itself in order
- 9.12 Consciousness and free will
- 9.13 Life is subtraction
- Chapter 10. The octet rule as arithmetic necessity
- 10.1 The structure imposed by nine digits
- 10.2 The octet theorem
- 10.3 The fold: concentric spheres
- 10.4 Historical precedents: Abegg and Linnett
- 10.5 The universal fold
- 10.6 The mirror cost
- 10.7 The Fibonacci cycle
- 10.8 The periodic table
- 10.9 The growth cycle
- 10.10 Conclusion of the Octet Rule
- Chapter 11. The end of integral calculus and the seven convergences of the half
- 11.1 The dead end of the continuum
- 11.2 The multiplication tables and the tipping at the centre
- 11.3 The pivot turns surface into volume
- 11.4 The volumetric projection and its correction
- 11.5 The seven convergences toward 1/2
- 11.6 Numerical validation and conclusion
- Chapter 12. The fractal law
- 12.1 Observation without the cause
- 12.2 The three conditions
- 12.3 The digital root, model iteration
- 12.4 The law and its proof
- 12.5 The iteration rule: bricks 2 and 3
- 12.6 The rhythm 24: six confirmations
- 12.7 Why Mandelbrot could not see the cause
- 12.8 Conclusion
- Chapter 13. The figure and the quantum
- 13.1 All geometry is projective
- 13.2 Tsirelson's bound and entanglement
- 13.3 Spin 1/2
- 13.4 Paradoxes as instrumental amputation
- 13.5 The administrative management of the irrational
- 13.6 The two original cycles and the projective form
- 13.7 Coda of Book III
- Appendix A: Table of statements
- Appendix B: Withdrawals and corrections
- B.1 Calculation corrections
- B.2 Status downgrades
- B.3 Passages not reproduced, in accordance with the rules of the corpus
- Appendix C: Merged sources
- Appendix D: Verifications redone