Gallery: One Pattern at a Time

Collection 1: “Digital Numbers”

The first Pyramid of Decimal Numbers counts from 1 to 440 and reveals the Top Right to Bottom Left Diagonal of Square Numbers. with Last Digits 1 4 9 6 5 6 9 4 1. The series repeats with ‘mirror effect’ at 5.

It highlights Multiples of 5 and 10.

The difference between cells equals 1. But between rows, the intervals are growing in even and regular steps.

2 Pyramid of DIGITAL Numbers

This Pyramid of Digital Numbers shows vertical, horizontal and diagonal differences between cells, rows and columns – a great improvement over the conventional Number Line, if we want to visualise relationships between Numbers.

3 Pyramid of Decimal Number SERIES

This Pyramid of Decimal Number Series shows numbers ending with the Decimal Digits 0 1 2 3 4 5 6 7 8 9 and how they appear once, twice, thrice or four times as Horizontal Series in rows.

4 EVEN Digital Numbers

Columns of Even Numbers, the Roots of Square Numbers and the Differences between Rows in addition to the Diagonal of Square Numbers are on display here.

Even Numbers end with Last Digits 0 2 4 6 8.

5 ODD Digital Numbers

Columns of Odd Numbers with Vertical Series lead inevitabily to the next image that unites the two complementary number qualities: odd and even.

Odd Numbers end with Last Digits 1 3 5 7 9.

6 ODD and EVEN Numbers

Columns of Odd and Even Numbers alternate and begin to display and intriguing interplay between odd 5, even 10 and odd and even Square Numbers.

Square Numbers end with Last Digits 0 1 4 9 6 5 6 9 4 1 0, i.e. they are ‘mirrored’ at 5.

Collection 2: “Last Digit Series”

7 Last Digits with 0 as Highlights

Decimal Numbers have 0, 10, 20… as their ‘unique identifiers’. Here we see which positions these take, as we look only at Last Digits.

8 Last Digits with 5 as Highlights

5 is Half of 10. Thus we see its multiples twice as often as the Last Digit 0.

How intriguing is the pattern 5 produces?

9 Last Digits with 0 and 5 as Highlights

0 and 5 together now! Interesting ‘spearheads’ pointing ‘forward’ to the right, as if the Number Universe could only be one-directional, especially if we want to consider it as a 3D template to model physical realities!…

10 Columns with ODD Repeated Last Digit SERIES

Last Digits in Odd Columns reveal yet more ‘whimsicalness’ of our Decimal Number System: repeated Vertical Series of 5 Last Digits in columns.

Diagonals show, mirrored at the central ‘Zero Column’, when the ends of Vertical Last Digit Series are seen together.

11 Columns with EVEN Repeated Last Digit SERIES
12 Odd and Even Repeated Last Digit Series
13 HORIZONTAL Odd and Even Series

Even Last Digits reveal the same principle of Repeated Vertical Series of Last Digit patterns – diagonally staggered and mirrored by the central even column.

I cannot imagine who will retain and infer what from seeing these Vertical Series in a pyramid. But it was a pleasure designing and perfecting them!

Maths is beautiful – especially when complementary qualities come together to rise to a higher level concept:-

  • odd and even digits and numbers;
  • horizontal and vertical series of digits or numbers;

resulting in The Diagonal as the next higher level concept:

  • diagonal Square Numbers and
  • diagonally staggered Limits of Vertical Series.

The following sequence presents one digit at a time.

Collection 3: Single “Last Digit Patterns”

14 The Pattern of Last Digit 1
15 The Pattern that Last Digit 3 produces in the Pyramid of Decimal Numbers
16 Last Digit 5 in the Context of all other Odd Last Digits
17 Last Digit 7 in its Pattern – as Unique as every other Odd Digit
18 Last Digit 9 and the Pattern it Produces
19 Altogether Now: Odd Last Digits amidst Columns of Even Numbers
20 And now Last Digits Only: EVEN in White Columns and ODD in ManyColoured Splendour – this is the ‘Patchwork with System’

Coll. 4: Composite “Last Digit Patterns”

21 From now on we see Combinations of Last Digits such as 1 and 3 here

22 Last Digits 1 and 7 in Close Vertical Proximity
23 Last Digits 1 and 9 strong in three Columns
24 Last Digits 3 and 7 also share Columns – just as 1 and 7
25 Last Digits 3 and 9 in yet more interesting Groupings
26 Last Digits 5 and 7 mainly in their own Vertical Series
27 Last Digits 5 and 9 with uniquely their own Vertical Series
28 Last Digits 1 7 and 9
29 Last Digits 3 7 and 9 in Decimal Pyramid
30 Last Digits 1 3 7 and 9, i.e. only Prime Digit 5 leaves gaps among white Even Columns
31 Back from ‘Conceptual Digital Heaven’ down to NumberLand: EVEN Decimal Numbers between ODD Last DIGITAL Numbers
32 Last Digits only: of Odd and Even Columns

Collection 5: “Digital Prime Diagonals”

33 Prime Digits and Prime Numbers
34 Prime Numbers grouped by Last Digits: 1 is grey and 3 7 9 are magenta
35 Primes grouped into 1 9 and Last Digits 3 7
36 Primes grouped into 1 7 9 and Last Digit 3
37 Primes Colour Coded by Last Digit 1 3 7 and 9
38 Primes as Last Digits 1 3 7 and 9
39 Primes as Last Digits 1 3 7 9 among Last Digits only
40 Primes on Four Parallel Diagonals
41 Last Digits of Primes on Four Parallel Diagonals
42 Last Digits of Primes on Four Parallel Diagonals among Last Digits

Collection 6: “Diagonal Sums”

43 ODD Diagonal Sums from Odd VERTICAL and Odd HORIZONTAL Summands
44 EVEN Diagonal Sums from Even VERTICAL and Even HORIZONTAL Summands
45 Triangle (Half a Matrix) of ODD Diagonal Sums from Odd Vertical and Even Horizontal Summands
46 Triangle of EVEN Diagonal Sums from Odd Summands in both Directions
47 Triangle of EVEN Sums from ODD Summands in both Directions
48 Triangle of ODD Sums on Parallel Diagonals
49 Chessboards of ODD and EVEN Sums

Collection 7: Primes as “Diagonal Sums”

50 Prime Numbers as Diagonal Sums of all possible Summands
51 Diagonal Odd Sums of Primes and Non-Primes
52 Diagonal Even Sums of Non-Primes
53 Primes and Non-Primes in “Diagonal Unison”

Coll. 8: Digital Matrices of SemiPrimes

54 Triangle of SemiPrimes
55 Last Digits of Triangle of SemiPrimes
56 Last Digits of Triangle of SemiPrimes with Symmetrical Double Diagonals
57 Last Digits of Triangle of SemiPrimes and the Orthogonal Nature of Prime Digit 5
58 Matrix of SemiPrimes
59 Last Digits of Matrix of SemiPrimes with Symmetrical Diagonals
60 Matrix of Semi- and Non-Primes LD

Collection 9: Factorisation of SemiPrimes

61 SemiPrimes as Products of Vertical and Horizontal Prime Factors
62 SemiPrimes with their Factors I
63 SemiPrimes with their Factors II
64 SemiPrimes with their Factors III – less colourful
65 Vertical Prime Twin of Primes and Factorised SemiPrimes
66 Unsorted SemiPrimes and their Prime Factors
67 Sorted SemiPrimes and their Prime Factors

Collection 10: “Digital Table” of Primes and SemiPrimes

68 Prime Numbers 1 – 1499 in “Digital Table” with Last Digits 1 3 7 9
69 Primes 3000 – 4493
70 Primes 4507 – 5987
71 Primes 4507 – 5987

Collection 11: “Digital Tables” for “SemiPrime Series”

72 SemiPrime Factor 7
73 SemiPrime Factor 11
74 SemiPrime Factor 13
75 Primes and Primes as Factors of SemiPrimes

Collection 12: Odd Products in “Number Planes”

76 Odd Numbers in a Number PLANE
76a Number PLANE with PRIME Axes for ODD Last Digits
76b Top Down and Bottom Up Pyramids Count 1 and Digit 0
76c Top Down Digit 0 and Bottom Up Count 1 Pyramids
76d Top Down Digit 0 and Bottom Up Count 1 Pyramids
76e Top Down Count 1 and Bottom Up Digit 0 Pyramids

Collection 13: “Number Planes” for “Prime Axes”

80 Compare and Contrast Decimal with “Digital” Numbers in a new Environment: a 4-fold Symmetrical Plane with Vertical and Horizontal Axes highlighting Prime Numbers
81 A Number Plane with four 10×10 Arrays of Sums
82 Four 20×20 Arrays illustrate Orthogonal and Diagonal Symmetries
83 Four 30×30 Arrays create Concentric Squares, Fully Symmetrical Diagonals (450) and “Prime Pairs” on “Horizontally Parallel Diagonals”
84 Four 40×40 Arrays display the Pattern that Connects Primes and Non-Primes, Diagonal lines and Orthogonal Axes
85 Orthogonal vs Diagonal Symmetries confluence into “OrthoDiagonal Dual Parallels”

Collection 14: “Orthogonal Products”

86 The OrthoDiagonal Nature of “Digital Sums”
87 The Orthogonal Nature of “Digital Sums” resulting in Multiples of 5 or 10
88 The Orthogonal Nature of “Digital Sums” of 5 and 10 with Prime Numbers
89 “Digital Sums” of Primes and Non-Primes and Multiples of 5 and 10 with Symmetrical Diagonals
90 The OrthoDiagonal Nature of “Digital Sums” – independent of Primes
91 The OrthoDiagonal Nature of “Digital Sums” and of this Pattern of Primes

Collection 15: Diagonal Sums of “Last Digits”

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