# 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.

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.

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.

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.

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.

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”

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

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?

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!…

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.

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.