“Pythagoras, son of Mnesarchos, a Samian, who was the first to call this matter by the name of philosophy, assumed as first principles the numbers and the symmetries existing in them, which he calls harmonies, and the elements compounded of both, that are called geometrical.” The Doxographists

Figure 1. Pascal’s Triangle from Blaise Pascal’s writings (Triangulo de Pascal en el escrito original de Pascal). Rows are designated rangs paralleles and columns rangs perpendiculaires. Labels have been added for the singular, linear, triangular, and tetrahedral continuous proportions. Each cell is equal to the sum of the preceding adjacent cells above (same column) and left (same row).

Figure 2. Point (1) and Void (0) shown rotated 90° relative to the plane of the page (condition will be defined as into and out-of the page plane going forward).

Figure 3. The continuous dimensional proportions of Pascal’s triangle begin with the contrasting continuous equal proportions of the tangential-void and the radial-point. Temporal proportions occupy rows and spatial proportions occupy columns.

Figure 4. The spatial and temporal increments corresponding to the continuous proportions of Figure 3 highlight the s/t dimensional duality of each differentiated numeric state within Pascal’s triangle.

Figure 5. The first four spatial and temporal dimensional perspectives (continuous proportions) are shown ordered by orthogonal dimensional constraints of point (1) and void (0). Void is shown summed with point, which thereby constrains consecutive continuous proportions to be at right angle to one another. The symbols used to define the six possible orthogonal dimensional constraint orientations are shown in the lower right of the figure.

Figure 6. The relative orientation of spatial and temporal geometries represented by the diagonal numeric species of Pascal’s Triangle. Dimensional constraint indicators are provided to show the anti-parallel relationship between spatial (column) and temporal (row) continuous proportion geometries (perspectives).

Figure 7. Spatial and temporal translational perspectives of the singular, linear, triangular, and tetrahedral dimensions are highlighted in yellow on the left and their respective geometries are shown on the right. The dimensional constraints that order these geometric perspectives are given as column and row headings.

Figure 8. The evolution of the spatial perspectives of the point and the corresponding evolution of numeric diversity: Follow along either “line of sight” of the spatially oriented point observing the dimensional geometries that emanate from the same centered origin. Numeric diversity (types of numbers) contained within each translational proportion is provided in the bottom left corner.

Figure 9. Dimensional number proof of the continuous space-time proportion equality: The figure shows that when an inter-dimensional term (perspective s-2) is added to the continuous arithmetic proportion equality, the relation is descriptive of all the translational perspectives of Pascal’s Triangle.

Figure 10. Triangular number modulus-9 relationship: A-planes begin with 1 and increment by 9n (1+9x0 = 1, 1+9x1=10, 10+9x2=28, 28+9x3=55). B-planes begin with 3 and increment by 9n+3. C-planes begin with 6 and increment by 9n+6. In this context n represents the nth triangle of a given species beginning with 0.

Figure 11. Cuboctahedral color-number polarity symmetry derived independently in two ways: either from a complementary cubic interpretation of published color symmetry or from the complementary cuboctahedral geometry of the 3-6 dipole highlighted through modulus-9 analysis of triangular number (Figure 10).

Figure 12. Complementary-cubic and complementary-octahedral dipole perspectives: counterclockwise cubic chromo-numeric polarization is shown on the left and clockwise on the right. Circle-dot means out of the plane of the page and circle-x means into the page as in previous figures.

Figure 13. Modulus-9 transforms Pascal’s Triangle into the dimensional number polarity field, which is then augmented using the complementary color-number affiliations defined in Chapter 5.

Figure 14. The Dimensional Polarity Field (27x27): The correlation between the number of dimensional constraints and modulus-9 invariance is most readily seen in the lower left and upper right quadrants where first-order translational analogs end with an incrementally increasing number of nines (1 to 8).

Figure 15. Extroverted (space-like) and introverted (time-like) dimensional constraints for spatial and temporal translational perspectives are shown relative to arithmetic proportion. Constraints are as defined in Chapter 3. The linear temporal perspective contains the second terms of all spatial translational perspectives and vice versa. These are the first terms to differentiate a given perspective from the singular. So I use the arithmetic proportion to show the relative constraint orientation of succeeding dimensions.

Figure 16. The complementary dimensionsal number polarity field versus Platonic Lambda proportions. Blue indicates field analogs and black indicates summed magnitudes of diagonals.

Figure 17. An expanded view of the dimensional polarity field showing the following overlapping analog periods: 3x3, 9x9, 27x27, and 81x81 (each is outlined in blue).

Figure 18. Scalar polarity field produced through modulus-9 reduction of a multiplication matrix. The 9x9 symmetry pattern (analog cell) repeats continuously throughout the scalar number field.

Figure 19. Complementary polarity transformations split into objective and subjective components: this analysis highlights the unique way each polarity transforms to its complement.

Figure 20. Primary, secondary, and tertiary phase distribution within the scalar polarity field.

Figure 21. Formation of the polarity field of continuous exponential proportions: The figure demonstrates that a 6x9 exponential polarity field repeats continuously throughout the arithmetically ordered whole number field. Tertiary polarities of the first and second analog are underlined for comparison since they differ between the first and all subsequent analogs.

Figure 22. Base-2 continuous geometrical proportion represented with 60° rotational transformations of 30°-60°-90° triangles about a central axis. The hypotenuse of a given triangle is the short side of the next larger triangle in the series. Equivalent color cube transformations are shown in the upper right to highlight alternation between primary and secondary phase hemispheres with 60° rotations about the 3-6 dipole axis.

Figure 23. Super-symmetric spatial intervals: accounting for incremental temporal constraint imposition aligns column-intervals. This is shown to correspond with cumulative temporal translational invariance in the right hand portion of the figure. The sums of columns are in base-2 geometric proportion, which is accompanied by clockwise complementary-cubic symmetry transformations as shown in Figure 22.

Figure 24. Complementary-cubic spatial and temporal rotational perspectives: Spatial and temporal constraints are shown just above their respective complementary-cubic chromo-numeric polarities. Cubic rotational transformations are shown above their respective 6-fold analog series with dashed arrows indicating the direction of rotation and solid arrows indicating the directions of polarization that correspond to the constraint orientations shown in the lower portion of the figure. To show the polarities of the back faces the front polarities are removed leaving white outlines.

Figure 25. The super-symmetric intervals of Pascal’s Triangle: Spatial/temporal rotational intervals are isolated by accounting for incremental imposition of temporal then spatial constraints (step 1 then step 2). These intervals correspond to the continuous golden proportions of the Fibonacci series (shown on the far right in modulus-9 chromo-numeric form). The spatial intervals along the top carry over from Figure 23.

Figure 26. Dimensional constraint rotation and accompanying modulus-9 polarity transformations are shown for spatial and temporal complementary-cubic proportions and for spatial/temporal and temporal/spatial continuous golden proportions (complementary-cuboctahedral). Temporal constraints are transposed from right-to-left (Figure 24) to left-to-right to enable side-by-side comparisons.

Figure 27. Spatial and temporal cubic 1-8 dipole rotation axes: Polarity analysis reveals 4-fold octahedral Fibonacci field segments. Asterisks indicate that the number 3 is paired with octahedral face polarity-6 and vice versa. Front views are indicated by circle-dot and back by circle-x. Arced dashed arrows represent rotation direction. Straight arrows represent polarization direction. Back views are shown from the perspective of the front so front arrows are shaded white and back arrows have an initial dash to indicate they are directed from the backside of the figure at an angle into the plane of the page.

Figure 28. Fibonacci series: Modulus-9 analysis shows that the 24-fold complementary numeric transformations of the continuous golden proportion are accompanied by super-symmetric cuboctahedral rotational transformations about the 1-8 spatial-temporal cubic dipole axis.

Figure 29. Spatial and temporal cubic 7-2 dipole rotation axes: Polarity analysis reveals 4-fold golden proportion field segments. Asterisks indicate that the number 3 is paired with octahedral face polarity-6 and vice versa. Front views are indicated by circle-dot and back by circle-x. Arced dashed arrows represent rotation direction. Straight arrows represent polarization direction. Back views are shown from the perspective of the front so front arrows are shaded white and back arrows have an initial dash to indicate they are directed from the backside of the figure at an angle into the plane of the page.

Figure 30. Polarity transformation analysis shows that rotation about the 7-2 spatial-temporal cubic dipole axis supports 24-fold complementary cuboctahedral geometric transformations associated with continuous golden proportions. The “Lucas” series (2, 1, 3, 4, 7, 11, and so on) begins at the sixth step of the 7-2 cubic dipole inversion sequence.

Figure 31. Spatial and temporal cubic 4-5 dipole rotation axes: Polarity analysis reveals 4-fold golden proportion field segments. Asterisks indicate that the number 3 is paired with octahedral face polarity-6 and vice versa. Front views are indicated by circle-dot and back by circle-x. Arced dashed arrows represent rotation direction. Straight arrows represent polarization direction. Back views are shown from the perspective of the front so front arrows are shaded white and back arrows have an initial dash to indicate they are directed from the backside of the figure at an angle into the plane of the page.

Figure 32. Polarity transformation analysis shows that rotation about the 4-5 spatial-temporal cubic dipole axis supports 24-fold complementary cuboctahedral geometric transformations associated with continuous golden proportions.

Figure 33. Interdependent 1-8 and 7-2 cuboctahedral dipole inversion fields: The neutral inversion centers of the 1-8 dipole field coincide with the octahedral centers of inversion between cubic poles of the 7-2 dipole field, as distinguished with dashed lines. A 4-fold pattern of polarity inversion transformations is highlighted using arced lines with arrows. Nested dipole inversions are shown with concentric arced lines.

Figure 34. The plot of the Ith roots of Lucas and Fibonacci series terms shows that the Lucas series (force)converges on the golden ratio and the Fibonacci series (energy)asymptotically approaches the golden ratio.

Figure 35. Interleaving cuboctahedral dipole fields of continuous golden proportion: “Derivative” fields are shown to begin with force and extend to the right. “Integral” fields are shown to begin with energy and extend to the left. This figure was generated by repeated summation of past and future intervals to form the present interval of the next dipole field to the right. This is shown graphically in the figure at interval zero where the past and future terms of the preceding proportion are summed to form the interval zero term of the succeeding proportion.

Figure 36. Field and anti-field symmetry: Spatial and spatial/temporal rotational perspectives of the origin of the energy field are shown juxtaposed with mirror temporal and temporal/spatial rotational perspectives of the origin of the anti-energy field. The dipole field of force (center) is the continuous golden proportion in harmony with sums of equivalent powers of the subjective temporal anti-base (yin) and the objective spatial base (yang). In contrast the fields of energy (right) and anti-energy (left) are the continuous golden proportions in harmony with difference-relationships between equivalent powers of base and anti-base, as shown.

Figure 37. The anti-field from the perspective of the field: the individual translational symmetry states are converted to complements, the CCW temporal complementary-cubic proportion becomes CW rotation of complements; and the complex complementary-cuboctahedral temporal/spatial rotational proportion orients to its spatial cubic pole (8).

Figure 38. Spatial/temporal oriented anti-energy and energy fields and associated rotational super-symmetries: full isotropic s-shell elements are shown as the culmination of spatial intervals; the first order symmetry field is shown to correspond to the theoretical limit of the periodic table; electrons and protons are shown corresponding to complementary-cuboctahedral dipole symmetries.

Figure 39. Spatial-Temporal Orientation and Translational Quantum Numbers: Left – Modulus-9 translational perspectives are shown with polygonal symbols corresponding to odd translational perspectives. Right – Valence shells, using spectroscopic notation, are shown corresponding to odd translational symmetries with spin-up corresponding to spatial orientation and spin-down corresponding to temporal orientation.

Figure 40. Quantum states shown in the governing 2:1 (spatial:temporal) constraint order with spatially oriented states over temporally oriented states (s/t).

Figure 41. Periodic Table of Atomic Elements derived from the first principals of Dimensional Number Theory. Lines are used to show how electrons add up to ultimately satisfy spatial then temporal translational symmetries.

Figure 42. Plot of first ionization energy versus atomic number (data courtesy of science.co.il). Half filled and filled shells are shown with polygon symbols indicative of their translational symmetries.

Figure 43. 3-4-5 triangle in-circle and ex-circle relationships: By constructing a rectangle with sides rv and rh atop the right triangle the sides of the triangle are found in terms of its in-radius and ex-radii. The radius of the right triangle’s hypotenuse ex-circle (rd) is found to be equal to the sum of the three other radii (r, rv, rh). And, the area of the constructed rectangle is found to be equal to the area of the right triangle.

Figure 44. Incircle/excircle diagram of the 5-12-13 Pythagorean triple triangle: This is the first triple triangle beyond the 3-4-5 triangle that has a odd delta equal to one (even delta increases from 2 to 8). Odd delta and even delta are shown as the difference between ex-circle radii and the in-circle radius. Notice that the geometric relationships established with Figure 43 remain valid.

Figure 45. Incircle/excircle diagram of the 15-8-17 Pythagorean triple triangle: The first triple triangle beyond the 3-4-5 triangle that has an even delta equal to two (the odd delta increases from 1 to 9). The geometric relationships established with Figures 43 and 44 remain valid. Note that the figure has been rotated by 90 degrees with respect to the previous two figures so that it will fit on the page with the same scale.

Figure 46. Near isosceles Pythagorean Triples (odd delta equals one): The first three triangles of the series are shown (3-4-5, 5-12-13, and 7-24-25). The continuous proportions of component lengths are provided in the accompanying table. odd and even deltas are shown equal to the length of their respective sides less the diameter of the incircle.

Figure 47. Delta Two Pythagorean Triples: The first three triangles of the series are shown (3-4-5, 15-8-17, and 35-12-37). The continuous proportions of component lengths are provided in the accompanying table. Odd and even deltas are shown equal to the length of their respective sides less the diameter of the incircle.

Figure 48. Primary and secondary prime phases: the primary phase consists of all integers that belong to the group 6n+1 and the secondary phase consists of all integers that belong to the group 6n-1, where n may be any integer. Prime numbers are underlined (primes courtesy of primes.utm.edu).

Figure 49. Numeric analysis of circle (rv + rh) and triangle (v + h) continuous prime-phase proportions for cases of delta one and delta two: For delta one, two thirds of numeric species belong to the primary prime phase and one third belong to the secondary prime phase. For delta two, one third of numeric species belong to the primary prime phase and two thirds belong to the secondary prime phase. In both cases 9-fold analog chromo-numeric transformations can be seen. Primes are underlined.

Figure 50a. Delta once circle prime and triangle prime continuous proportion divisor analysis: Terms divisible by a given preceding term are separated into two series and are color-coded. The terms of a given separated series occur at a frequency equal to the reciprocal of the divisor term, for instance every 5th term is divisible by 5. The two series associated with the given term are separated by triangular and linear offsets as shown.

Figure 50b. Delta two circle prime and triangle prime continuous proportion divisor analysis: Terms divisible by a given preceding term are separated into two series and are color-coded. The terms of a given separated series occur at a frequency equal to the reciprocal of the divisor term. The two series associated with the given term are separated by triangular and linear offsets that are oriented opposite to those of the delta one case.

Figure 51. Isolated sub-series continuous proportion of the circle and triangle prime series terms: Each sub-series is affiliated with their respective prime divisors (5 and 7) through the ip^2 continuous triangular proportion. Modulus-9 symmetry and the continued proportion of differences between terms indicate that series-1/series-2 pairs are opposite directional components of the same continuous proportion.

Figure 52. Merged ip^2 continuous proportion of first two terms for both circle and triangle prime series for fixed delta one and two.

Figure 53. Singular triangular/linear interval perspective: Upper-left the 2:1 (spatial:temporal) constraint asymmetry aligns the triangular-spatial and linear-temporal perspectives of the point to the same spatial/temporal interval. Lower-left this singular triangular/linear interval is shown as the culminating state of the primordial-order field analog. To the right the orthogonal triangular (spatial) and linear (temporal) perspectives of the 3-4-5 triangle combine to form the spatial/temporal proportions the diagonal.

Figure 54. Prime phase analysis of the Fibonacci and Lucas series: Primary prime phase terms are highlighted with bold-white and secondary with bold-black. Actual prime terms are underlined. The analysis shows a clear prime-phase pattern for both cuboctahedral dipole fields. This pattern suggests that the prime phase mediates inversion between spatial and temporal polarization.

Appendix A1. Triples tables: fixed odd delta (1, 9, 25) on the left and fixed even delta (2, 8, 18) on the right, the first 18 in-radii for each series is provided, primes underlined.

Appendix A2. Triples tables: fixed odd delta (49, 81, 121) on the left and fixed even delta (32, 50, 72) on the right, the first 18 in-radii for each series is provided, primes underlined.

Appendix A3. Triples tables: fixed odd delta (169, 225, 289) on the left and fixed even delta (98, 128, 162) on the right, the first 18 in-radii for each series is provided, primes underlined.

Appendix B1. Spatially polarized Complementary Cuboctahedron template: For best results color photocopy template on to card stock. Cut out and then crease along edges of adjoining faces and assembly tabs. Then glue and secure tabs until adhesive dries.

Appendix B2. Temporally polarized Complementary Cuboctahedron template: For best results color photocopy template on to card stock. Cut out and then crease along edges of adjoining faces and assembly tabs. Then glue and secure tabs until adhesive dries.