Circlemath

Its Patterns and Idea

Circlemath is the subjectivity of mathematics in general, the form it takes in the mind before it ever appears in the world. As the human brain underpins human language, whatever national or dialectal form this may take, the human brain underpins the use of mathematics, and the study of this subjectivity or underpinning converges immediately upon the process of thinking. The study of one entails a study of the other; the best way to introduce the subject however is to look at some figures.

Ordinary math teaches that 9*9=81. This is a useful fact, much quicker than counting nine lots of nine pebbles into a jar and then counting the total. In the following discussion, ‘OM’ will stand for ordinary math and ‘CM’ for circlemath. The two disciplines belong together, but we will take them separately for a start in order to see their difference. Only later will we see their unity, at which point new subjects will begin to take shape.

***

CM looks at the 9*9=81 statement, and adds that, besides counting, you can also see its logic in the fact that the answer, 81, adds, 8+1=9.

OM gently corrects him, saying, “The ‘8’ in 81 is eight tens, and eight tens, plus 1 is 81, not ‘9’. You cannot add a power digit (the 8) and a unit’s digit (the 1) as if they were of equal rank.”

CM “You can in circlemath, because the circles manage the units and powers for you. They preserve these relations in the calculations. We can calculate in given bases and across them as well. Circlemath gives us this extra degree of freedom.”

OM “What do you mean by a degree of freedom?”

CM “When we count or calculate in ordinary math we stay in a chosen base. We can move up or down its extent from 0 to infinity. This is one degree of freedom. In circle math, we can stay in the chosen base, but we can also move the operation sideways into other bases. The straight counting line of ordinary math is like a tightrope. We cannot step off it. Circlemath allows lateral movement transbase, so calculation is less like a tightrope and more like a bridge. This is a second degree of freedom. The number pattern in 81= 8+1=9 belongs to this transbase interoperability. Conventional math is decimal centered, where decimal means ‘fingers ten’.”

OM “Mathematics excludes contingent relations. The 8 1 relation in ‘81’ is not universal. If the relation between the 8 and 1 in 9*9=81 were universal, it would also be present between the 6 and 4 in 8*8=64. It is not. We cannot therefore allow your observation. Mathematics is rigorous and precise and proves every step of the way.”

CM “To be precise you would have to say that the 8 1 relation in question is not universal in terms of conventional straight line mathematics, but it is so in terms of circlemath, which is even more rigorous, for its algorithms are required to apply unchanged in any and every base. Furthermore, it requires no proof for it interfaces directly with our understanding, the ground upon which all proof in ordinary math stands. But let me expand my initial statement on the 81, 8+1 relation and then attend to the 8*8=64 question.

The 8 and 1 (in 9*9=81), sum to 9. The 7 and 2 in 9*8=72 sum to 9. The 6 and 3 in 9*7=63 sum to 9, and so on for 9*6=54 etc. The pattern is that of the nine direction in ten circle.

Figure 1

All the ‘nine times’ answers are represented: 09 18 27 36 45 54 63 72 81 and 90. Note that the constant multiplier, 9, is -1 in ten circle.”

OM “The circle offers a multiple conformity, but it is not universal, and this excludes it from mathematics. Now, my question concerns 8*8=64.”

CM “I will answer both your observations, but take 8*8 first. Ten circle does not show the pattern for 8*8 directly, but it is there, hidden. It needs an algorithm to bring it out, but rather than go into that here (it would take us into the bowels of circlemath), glance at the following sequence:

8*8=64, = 6+4, = 10, = 1+0, = 1

8*7=56, = 5+6, = 11, = 1+1, = 2

8*6=48, = 4+8, = 12, = 1+2, = 3

8*5=40, = 4+0, = 04, = 0+4, = 4

8*4=32, = 3+2, = 05, = 0+5, = 5

It is the same adding process we used for 9. You cannot see the leopard, but you can see its spots. The answers, 1, 2, 3, 4, 5… clearly indicate a pattern!

To move on to your other point, that of universality, there are two levels. The first is deep, the second shallow. Compare, if you will, the difference between subatomic and atomic physics. One is substance, internal, the other, surface, external.

Reverting to the 9*9=81 example, we can take this into other bases. ‘9’ in ten circle is ‑1. Choosing base eight because we can easily secure its answers on an octal calculator, ‑1 in eight circle is 7. The relation transfers therefore as 7*7=61. Again we find that the answer, 61, adds, 6+1=7. The following diagram shows this.

Figure 2

The ‘add to 7’ lines (fig. 2), indicate the ‘7-direction’ in eight circle, and there is the 7*7=61 answer sitting there, along with the others, 7*6=52 etc. Just read them off. Alternatively, tap out 7*7 on an octal calculator. It gives 61.

These circle-straight relations in adding subtracting multiplying and dividing repeat in every base-founding circle to infinity and constitute the beginning of a transbase uniformity that brings all number, in-base and trans-base, into a thought-amenable system. Beyond this, mathematics becomes neurology.

Universality

In contrast, ordinary math is universal only within a given base (usually decimal). Circle math, whose algorithmic steps are more fine-grained, is universal within a given base, but also, simultaneously across all bases. This transbase uniformity is lost, the moment we break the circles open, when they, as ideal (perfect) circles, transform into equally ideal Euclidean straights.[i]

We have 7*7=49, in decimal, and we accept that, but it is not true in other bases. We have seen that 7*7=61 in octal. The fact that a base-changing rigmarole interrupts calculation goes back to the rote tables, which substitute authority and memory for due process and understanding. ‘Due process and understanding’ here, is of course, recognition of the primacy and use of the circles. Calculators allow us to jump from base to base, but it takes a keystroke to switch to octal, and what if we want base 13, or base 3013 instead? Calculators that work indifferently across all bases do not exist, except in the human head, and we call their output thinking. Transbase calculation without circlemath is as impossible as the Internet would be, if every computer came with a unique brand of software.

Calculators do not diminish the importance of our inherent mathematical ability. In activities such as shopping surveying weighing and evaluating, we still figure using remembered ‘times tables’. It is quicker to know that 7*8 decimal is 56 than to reach for a calculator and tap it out. However, this inculcated dependence upon rote memory prevents us from recognizing the transcendent significance that attaches to the transbase domain. Since it is only dealing with facts it is not too extreme to say that for centuries educators have presented a one-sided view of mathematics from behind a wall of precision and authority, which from its vantage point in the schoolroom has penetrated the whole of knowledge, scientific and social alike. The error has reached out beyond the confines of mathematics, becoming the focal point of a logjam in all knowledge.

Circlemath provides a logical approach to 7*8=56. Add the 7 to the 8 in ten circle. It comes to 5, the power digit of the answer. Multiply the 7 by 8 (or using complements, 3 by 2). This gives us 6, the unit’s digit. This 7+8=5 and 7*8=6, read from line patterns in ten circle, allows children to construct the tables, and take them just as easily into the twenties and thirties, as well as jump from one base into another.

Math is a form of reasoning, and as in reason, we have the choice of many pathways to the answer. We can reach, pass and return to a desired point in a mental process that fills the gap between counting and calculation.

Within the circle of a single base (namely ten circle, which gave us 7+8=5 and 7*8=6), circlemath arrives at its tens and units digits independently. There are no carries. Normally we would say, 7+8=15 and 7*8=56, but the circles go straight from single-digit results to final answers, this one being to 56 in a hundred circle (the actual arithmetic for the power digit, performed by the circles, is 100 – (30+20).

Counting is a logical process, powered by an in-mind mechanism. Jumping to rote-learned answers, ordinary mathematics leaves logic behind and becomes authoritarian. Children are required to learn their tables, some 400 equations in adding subtracting multiplying and dividing. Circlemath never parts company with the logical process, which is geometrical and intelligible. Circles and lines give circle results, which then combine to give final answers.

Its algorithms change form as the numerical aspects of a calculation alter, but they apply unchanged to matching formations from base to base. This is their universality, or second-degree freedom. There is no substitute however, for knowing how they apply and change from calculation to calculation and base to base. Circlemath, in this respect, is like learning a language, or to play a musical instrument.

Our sum, 9*9 in decimal, became 7*7 in octal. We can now take it into hexadecimal, where it becomes F*F in the context of ‘16’ circle (fig. 3c below).[ii] The calculator tells us that F*F=E1, and in figure 3c we can see E1 sitting there. E is 14 and F is 15, so E+1=F. This confirms the pattern, which holds in all base circles ‘to infinity’.

Incidentally, we might as well try to get to the moon in a wheelbarrow, as become proficient in transbase work using bases that substitute alphabet letters for numbers. For a start, it is opaque, and for the higher bases, it would require millions of unique symbols, which simply do not exist.

Notice that the numbers (15 by 15 in this case)[iii] determine the calculations, but the circles shape them, and this is so in every base. Transbase calculation is possible because universal algorithms exist for all number relations, not just to the examples used here. Universality means without exception in all bases to infinity.

Figure 3 shows ‘16’ circle alongside an outline of the brain. Both exhibit the same set of numbers, 0 to F. The circle shows the math logical relations. Fig. 3b indicates that cells scattered throughout the brain project a mathematically ordered determination.

Figure 3

The brain, in generating reasoned behavior, pays attention only to functional results. The order we see in mathematics does not show in the brain's neural organization, but it is there. Circlemath, which mediates between the brain and the math of the world, is transbase and infinitely rich in meaningful pattern. Figure 3 intends to convey no more than an impression of relation between numerical pattern and brain circuitry. The actual relation is a second step, for which see, Circlemath and Neurology, and beyond this, ATOM, a Theory of Mind.

Extend the example 9*9=81, to 9*9*9=729. Because 729 groups in circle idiom as [72] 9, and 72+9=81, we immediately see the answer as an expansion of the 9*9=81 form. We can now predict answers for the similar calculation in sequential bases. The list below extends from binary to F*F*F.

Calculators will usually give answers in octal, decimal or hexadecimal, but nothing in between, because in ordinary math there is no call for answers in bases 9 11 12 13 14 15. This is, of course, a weakness, not strength. Ordinary math bows out if asked to give answers in bases running into the millions and billions (whose numbers do not have to be rounded). This is sufficient proof that universal patterns exist beyond the reach of standard math, and precisely here, we cross the threshold to circlemath, which is at home in mutibase calculation.

The list below presents the transbase relation in a mathematically accountable form. You can accept that it is correct and skip-read to the next heading, ‘The Larger Story’, or you can systematically check every step.

Because it is quite difficult by ordinary methods to confirm the given answers in the different bases, even though the numbers involved are quite small, doing the math will impress those who make the effort, that by pattern alone circle procedure can lay out answers ‘to infinity’.

The pattern is truly transbase. To fully appreciate this is to carry the conviction that mathematics, as currently taught, is but a single facet of a greater universality.

Calculation in Transbase Perspective

1 * 1 * 1 in binary = 001 (no prediction needed, packing with left 0’s is optional)

2 * 2 * 2 in base 3 = 022 (the multiplier is always the units digit of the answer)

3 * 3 * 3 in base 4 = 123 (2 ends the grouping, [12]3 in 123)

4 * 4 * 4 in base 5 = 224 (the group adds to the units digit, here 2+2=4)

5 * 5 * 5 in base 6 = 325 (the units digit of the answer is 5, and 3+2=5)

6 * 6 * 6 in base 7 = 426 (we have the pattern. Is the answer correct?)

7 * 7 * 7 in octal = 527 (check the answer with an octal calculator)

8 * 8 * 8 in base 9 = 628

9 * 9 * 9 in decimal = 729 (this is our familiar base, so it is easy to confirm)

A * A * A in base 11 = 82A

B * B * B in base 12 = 92B

C * C * C in base 13 = A2C

D * D * D in base 14 = B2D

E * E * E in base 15 = C2E

F * F * F in hex = D2F (‘16’ circle gives the answer; a calculator confirms it)

The Larger Story

OM “The pattern you are discussing is only a modification of ‘casting out nines’. There is nothing new in this. It exists in hundreds of school-level arithmetic books. Your earlier reference to 7+8=5 and 7*8=6 in ten circle, giving 7*8=56 in a larger circle, is similarly an adaptation of Gypsy multiplying. These methods are hundreds, if not thousands of years old.

CM “Ah... we are beginning to cross the bridge! What you say is true, but these existed only as number gimmicks. The parent universal system remained hidden. We need this deeper comprehension in order to fit math into the wider picture of mind and life, of which it is a part, alongside and integrated with the other great disciplines, theology philosophy and science, not as an island or walled fortress.

Your earlier rejection of the 8+1=9 relation (in 9*9=81) illuminates the fact that the in-mind depth of ‘casting out nines’ has not been known, for we cannot cast out eights, sevens, sixes… Gypsy multiplying is an aid to some aspect of the tables, but not to others. Mathematics will come forward as a science when it is no longer taught as a blind mechanism built upon a platform of rote-learned tables.

The very idea of ‘casting out nines’ is wretched. It is true only in decimal. In base nine it is ‘casting out eights’; in base eight it is ‘casting out sevens’ (as we saw), and in base sixteen (hex) it is ‘casting out F's’ or fifteens. We can cast out any number, but we must shift base to do so. The method does not hinge upon the nine at all, but –1 in a circular framework.

To see this is to begin to make the transition to a greater mathematical world. It also applies in Gypsy multiplying, which is also a patch of the universal ‘showing through’ within the framework of accepted rote-based straight-line math. We must go beyond this rote and rule of thumb to the foundation that sustains ordinary math and thought as well, transforming in the process into the greater subject of mind.

Circularity unites numerical geometrical and algebraic expression in the universal algorithmic system that is the motor or mechanism in abstract thinking. To divine how math and thought come together in the depths of the brain to form the mind is a daunting task, but necessary, given the way modern knowledge is developing. We must attend to it, in order to keep abreast of the potential unfolding in technology, a potential otherwise as dangerous as a loaded gun in the hands of a child.

The numerical patterns shown, within selected bases as well as transbase, point to the existence of a groundswell of universal relation that underpins all mathematics, coincident with the nature of mind in its neural bed. This essay set out to show no more, so it is time to end. The next step belongs to neurology, which, taking mind back to its biological foundations will prove to be the link, unifying all knowledge, a unity that will find its way into the wider reaches of our many faceted world.

Top