by H. PeterAleff |

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Figure 1: Some square "Sieves of Eratosthenes" with primes and other special numbers marked Figure 2: Rectangular arrays four and six units wide, and the distribution of primes into only two of their columns
1.3.1. Numerical versus physical universes 1.3.2. The ongoing search for prime laws
Figure 4: Polygonal numbers as dot figures, and some simple visual proofs of their relationships 1.4.1. Gnomons and triangle chemistry 1.4.2. 1.4.3. The triangular- number "pyramid" Figure 5: The top of the triangular- number stairway, and its alignments of primes Figure 6: The same number- line segments as in Figure 5, but stacked symmetrically 1.4.4. The square- to- square "pyramid" Figure 7: The apex of the square- to- square "pyramid", and its strings of primes
Table 1: Angles of repetition for columns and edge parallels
Figure 8: Apex of pentagonal- number pyramid, with prime- solid diagonal strings Figure 9: Apex of hexagonal- number pyramid with diagonals and prime- free columns Figure 10: Apex of heptagonal- number pyramid, and its prime- solid diagonals Figure 11: Apex of octagonal- number pyramid, with strings of twin prime centers Figure 12: Apex of nonagonal- number pyramid
Figure 13: Patterns in the gaps between the primes of the triangular- number pyramid column at 7. Earlier column members return in a double cascade as factors in later ones. Figure 14: Gaps between primes in the Euler Column under five, their composition from earlier primes, and their progression of "guest factors"
Table 2: Curious connections between the triangular- number and square- to- square number pyramids
Figure 15: The upper clock- face of the "number calendar" centered in the square- to- square number- pyramid array Figure 16: The lower clock- face of the "number calendar"
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