origins

Arithmetic Foundations

January 9, 20108 min read

Original blue and orange circular arithmetic chart, showing the staggered boundaries between colored bands on nine radials. — Palindromes in every direction. The complement map, before it had a name.

An origin essay from January 2010, revisited in September 2026.

Add nine to 91 and you get 100. The two numbers have the same remainder when divided by nine, but their decimal digit sums differ. The digits of 91 add to ten. The digits of 100 add to one.

These drawings keep both facts in view. The long table records the arithmetic entry by entry. The circle puts the sequences beside one another, where their alignments and offsets become easier to see. Moving between the two became one of the most useful habits I took from this early work.

I was still learning number theory. Making a picture gave me something definite to investigate, and keeping the table underneath it gave me a way to check. Some of my strongest insights began with that simple exchange.

In the circular view, follow a spoke outward, then trace a band across the neighboring spokes. The first path reveals a sequence of returns. The second reveals the shifts between them.

Original circular arithmetic chart with nine spokes, blue and orange bands, nine/zero at noon and four and five below. — The whole chart. Follow one spoke outward to see the bands widen and narrow; then look across neighboring spokes to see how their boundaries shift. The original yin and yang labels remain in the artwork.

Two coordinates

The drawing arranges the integers from 1 through 2016 on nine spokes. Each number goes on the spoke given by its remainder after division by nine. One, ten, nineteen, and twenty-eight share a spoke. Nine, eighteen, and twenty-seven share the zero/nine spoke. Each step outward adds nine.

The second coordinate comes from adding the decimal digits. Keep that sum before reducing it any further.

For 19, the sum is 10. For 28, it is also 10. For 100, it is 1. All three belong to the same spoke. The digit sum distinguishes numbers whose remainder gives them the same address.

In the rectangular table, the columns record the original number, its digits written as an addition, their sum under B10, and the remainder under M9. The colors come from the digit sum. Blue covers 0 through 8, gold covers 9 through 17, blue returns for 18 through 26, and gold again for 27 through 35. The circular drawing uses orange for the gold regions.

See the original rectangular table overview

Original blue and gold arithmetic table, showing the integer, its decimal digit sum, and its modulo-nine remainder in parallel radial panels. — The table laid flat. The zero/nine class appears at both edges, so there are ten panels for nine classes. A full-resolution reading view is available at the end of this essay.

The change from 91 to 100 now has a visible consequence. The spoke stays fixed, while the color changes from gold to blue. A decimal carry moves the number into a different band. Repeating the same addition builds a much richer picture than the fixed remainder alone would suggest.

A palindrome along the radial

Begin with the zero/nine column. Leave the initial zero aside and start at nine.

The ten entries from 9 through 90 all have digit sum 9, forming a gold run. The next entry, 99, has digit sum 18. It makes a blue run of one.

After that come nine gold entries, from 108 through 180, followed by two blue entries, 189 and 198. Then eight gold entries and three blue ones. One run shortens as the other lengthens.

Gold run	Number of entries	Following blue run	Number of entries
9 to 90	10	99	1
108 to 180	9	189 to 198	2
207 to 270	8	279 to 297	3
306 to 360	7	369 to 396	4
405 to 450	6	459 to 495	5

Continue through 990 and read the successive run lengths.

10, 1, 9, 2, 8, 3, 7, 4, 6, 5,
5, 6, 4, 7, 3, 8, 2, 9, 1, 10.

Read the list backward and it stays the same. The integers keep increasing while the run lengths reverse their sequence.

This is the radial palindrome in a form we can count. Across the whole stretch there are 55 gold entries and 55 blue entries. Gold runs shorten from ten entries to one as blue runs lengthen from one to ten. Reversing the window also exchanges the colors.

This particular palindrome runs from 9 through 990. The full table continues through larger carries, letting us inspect how the pattern changes beyond those endpoints.

Mirrored endings

The one-spoke in the five hundreds gives another way to see a return. Begin with its first six entries.

505, 514, 523, 532, 541, 550.

Keep the leading five fixed and watch the endings.

05, 14, 23 | 32, 41, 50.

The last three reverse the digits of the first three, in the opposite order. Every entry has digit sum 10. Adding nine advances the tens digit while reducing the units digit. The entries walk through the little mirror one step at a time.

The next five entries are 559, 568, 577, 586, 595. Their endings are 59, 68, 77, 86, 95, with 77 at the middle. Their digit sum is 19, so they occupy the next color band.

There are now two related patterns to follow. The endings reverse inside a run, while the runs themselves change length. Reducing everything to the spoke number would erase both. Keeping the intermediate arithmetic makes them visible.

The turn between radials

Lay the columns around the circle. The color changes occur at different distances from the center because neighboring spokes are at different stages of their runs. Trace a boundary across the spokes and those offsets give it a twist.

The interlaced spirals bring the two directions together. Along a radial, we can follow the run-length palindrome. Across radials, we see the displacement between those patterns. I later used torsion for the coupling between the two directions.

Original annotated detail of the arithmetic spiral, with black and red arrows following changes in the bands beside the integer labels. — The annotated detail follows the changing widths of the bands. Compare the boundary along one radial with its position on the next. The integer labels keep the drawing tied to the calculation.

I could follow a shape into the table, locate the entries that made it, and return to the circle with a more precise question. That was a large part of learning to work with these images. The drawing suggested a relationship. The arithmetic gave me somewhere to put my finger.

Closer views

The original magnifications make that journey in small steps. At a distance, the broad bands stand out. Closer to the center, the numbers become legible and we can check the entries on either side of a boundary.

Original eight-times magnification of the circular arithmetic chart, with broad interlaced bands and numbered radials. — At 8×, follow the changing width of a band across several spokes.

Original thirty-two-times view near the center, with individual values including 19, 28, 37, and 46 visible on the one-spoke. — At 32×, read the entries themselves. Nineteen, twenty-eight, thirty-seven, and forty-six share a spoke and digit sum 10.

See the original 2×, 4×, 6×, and 16× magnifications

Original two-times magnification showing the larger blue and orange band arrangement. — At 2×, the larger arrangement is visible, with individual entries still densely packed.

Original four-times magnification of the arithmetic chart. — At 4×, the radial runs and their offsets become easier to separate.

Original six-times magnification with numbered spokes and changing color-band widths. — At 6×, compare a band boundary on one spoke with its position on the next.

Original sixteen-times magnification, showing readable arithmetic near the center. — At 16×, the individual entries begin to emerge from the band pattern.

All six enlargements show the same construction. The sequence keeps the larger arrangement in view as the individual entries become readable.

Multiplication paths and primes

The multiplication drawings fit the same nine positions. Their six-position doubling path and the triangle through three, six, and nine can be laid directly over the arithmetic chart.

The earlier glyph on the arithmetic chart. Both use the same nine positions, making the radial bands and the multiplication paths available in one view.

I also put the primes back into the picture, as I had in The Circle of Nine. This time they sat against the full pattern of digit sums and carries.

Original prime overlay on the full arithmetic circle, with prime entries highlighted against pale background bands. — The prime overlay. Apart from three itself, primes occupy the six spokes that are not multiples of three.

Original magnified prime overlay with individual integer labels visible along the radials. — The closer view keeps the surrounding composite numbers visible too. A cluster can be inspected in its arithmetic neighborhood.

I wanted to examine the primes in relation to the bands and their joins. Keeping the composite numbers visible gave me the surrounding arithmetic as a reference. I could choose a stretch of the picture, inspect its entries, and compare it with another stretch. The overlay gave that investigation a place to begin.

Balance and energy

These pictures helped me distinguish radial palindromic energy from torsional energy. The table made the return along a radial countable. The circle showed the relative shifts between radials. Keeping both in view became essential to my thinking about balance.

A balance can conceal activity. A step of plus one followed by a step of minus one has no net change. Square the two steps and their contributions add to two. The two totals answer different questions about the same pair of steps. These drawings made me attentive to that distinction long before I could formulate it cleanly.

The same line of thought kept bringing me back to two thirds and one third, and eventually to the cubic law for digit-collision energy. In that later theorem, equal collision ratios contribute one third of the leading cubic mass. Unequal coprime ratios contribute two thirds. The larger contribution comes from the unequal ratios. These proportions emerge as the prime base grows, with finite corrections accounted for separately.

Those fractions measure energy in the later collision calculation. The 55-and-55 count above measures entries in one stretch of the early drawing. The connection is in the questions the pictures helped me ask about return, relative shift, and the activity that survives a cancellation.

I still use these drawings. I can follow a column, find the carry that changes its color, then return to the circle and see how the local change fits into the larger pattern. There is a practical advantage to keeping the arithmetic underneath a picture. Years later, I can still pick a column and begin again.

The full table

The full-resolution table below is meant to be read. Open it, use the zoom controls, and move across or down the image. Start with the MOD9 Web Radial 0/9 panel at the left. Find the gold run from 9 to 90, the single blue 99, and the nine gold entries that follow. Then compare the neighboring one-spoke.

The complete Tables of Foundational Arithmetic through 2016, at 2000 by 3093 pixels, with all integers, digit sums, remainders, and blue/gold runs. — The original full-resolution table. Open the table and zoom in. The source image and its attribution are preserved.