Steve Whealton
Ternary Arithmetic
Here, we will focus on the numbers from 0 up to 242, expressed in ternary, or base three, arithmetic. Whenever necessary, we will add leading zeroes to the ternary patterns that we are working with.
So 0 will be expressed as "00000," and 242 will be known as "22222," and all of the numbers in between will have five digits as well, all of being "0," "1," or "2."
Given all of this, our transforms can be seen in two rather different ways at once. On the one hand, we are transforming numbers between 0 and 242 (inclusive) into (usually) other numbers between 0 and 242 (also inclusive). On the other hand, we will also be transforming one pattern of the form, "XXXXX," into another pattern of the form, "XXXXX," where in each case all five "X's" can be anything from "0" to "2."
Sometimes we will think numerically first and then look at the "XXXXX" patterns. But most of the time we will be begin with patterns and only after planning some transform or other in terms of pattern will we look at the numerical side of things.
We will also insist that all our transforms be one-to-one. This means that each of the 243 numbers between 0 and 242 (inclusive) will be transformed into a unique number between 0 and 242 (equally inclusive). No two numbers will end up in the same place, and no numbers will fail to be the end point of some transformation process.
One could line up all 243 numbers in a column on the left and then line them all up again in a column on the right. Then, the transform could be seen as a set of 243 lines, each one connecting one number in the left-hand column with a number in the right-hand column. No number on the left or the right will have more or less than one line linked with it.
The laws of combinatorics insist that transforms of this kind will, if applied over and over, eventually recreate the same pattern that was present when the transformations began. But the number of transformations necessary before such a return must take place can be very large indeed. The transforms we will be working with will have fairly small cycles, and in every case we will talk about these cycles.
The simplest really useful transform is simple subtraction from 242. This transformation reverses the order of the numbers. There is a halfway-number, 121, which in ternary is "11111." Unique among all the integers within our chosen range, this one survives subtraction from 242 unchanged. 242 - 121 = 121, after all. Similarly, 22222 - 11111 = 11111.
All other numbers are transformed into numbers other than themselves.
Is this transformation one-to-one?
Intuition insists strongly that this be so. Subtracting any number from 242 produces a unique result.
When the numbers between 0 and 242 are linked (as they usually have been, when I have been working with transformations) with the amount of light present in a pixel on a computer monitor screen, then this transformation can easily be seen as equivalent to taking the negative of an image. Dark areas become light; reds become green, and only a maximally neutral gray will remain the same.
The table below gives a few typical (!?) numbers and shows their reversed form:
191 |
21002 |
rev |
01220 |
51 |
25 |
00221 |
rev |
22001 |
217 |
121 |
11111 |
rev |
11111 |
121 |
0 |
00000 |
rev |
22222 |
242 |
242 |
22222 |
rev |
00000 |
0 |
194 |
21012 |
rev |
01210 |
48 |
89 |
10022 |
rev |
12200 |
153 |
225 |
22100 |
rev |
00122 |
17 |
65 |
02102 |
rev |
20120 |
177 |
The next transform that we will look at is left-right inversion. In some ways, it is like the transformation outlined just above. They are alike in that each one transforms its number into an opposite of some kind. In this case, we will look at the ternary patterns and turn them around, left to right.
191 |
21002 |
lri |
20012 |
167 |
25 |
00221 |
lri |
12200 |
126 |
121 |
11111 |
lri |
11111 |
121 |
0 |
00000 |
lri |
00000 |
0 |
242 |
22222 |
lri |
22222 |
242 |
194 |
21012 |
lri |
21012 |
194 |
89 |
10022 |
lri |
22001 |
217 |
225 |
22100 |
lri |
00122 |
17 |
65 |
02102 |
lri |
20120 |
177 |
Is this process one-to-one? Is it also revsible? Looking just at the 5-digit patterns, intuition says that any pattern, when reversed from right to left, will become a single, unique, pattern. Do all of these transformed patterns have to be within the required range? Again, intuition says that because all 5 digits still have to be "0," "1," or "2," then there is no way that a transformed number can stray beyond the required bounds.
Why not try mixing the two transforms, next? To do it properly, we should apply the two first in one order and then, in the other:
First, we perform subtraction from 242, which we will abbreviate "rev," and then we will perform the left-right inversion, which we will abbreviate "lri."
191 |
21002 |
rev |
01220 |
lri |
02210 |
75 |
25 |
00221 |
rev |
22001 |
lri |
10022 |
89 |
121 |
11111 |
rev |
11111 |
lri |
11111 |
121 |
0 |
00000 |
rev |
22222 |
lri |
22222 |
242 |
242 |
22222 |
rev |
00000 |
lri |
00000 |
0 |
194 |
21012 |
rev |
01210 |
lri |
01210 |
48 |
89 |
10022 |
rev |
12200 |
lri |
00221 |
25 |
225 |
22100 |
rev |
00122 |
lri |
22100 |
225 |
65 |
02102 |
rev |
20120 |
lri |
02102 |
65 |
Next, we put lri first, and then subtraction from 242:
191 |
21002 |
lri |
20012 |
rev |
02210 |
75 |
25 |
00221 |
lri |
12200 |
rev |
10022 |
89 |
121 |
11111 |
lri |
11111 |
rev |
11111 |
121 |
0 |
00000 |
lri |
00000 |
rev |
22222 |
242 |
242 |
22222 |
lri |
22222 |
rev |
00000 |
0 |
194 |
21012 |
lri |
21012 |
rev |
01210 |
48 |
89 |
10022 |
lri |
22001 |
rev |
00221 |
25 |
225 |
22100 |
lri |
00122 |
rev |
22100 |
225 |
65 |
02102 |
lri |
20120 |
rev |
02102 |
65 |
Clearly, 9 examples do not constitute a proof. Yet a proof is possible. One can look upon these transforms as fodder for the creation of a group. The identity transform is one member, rev is another, lri is another, and our combo is thr fourth.
It is interesting to look closely at which numbers survive, unscathed, under which transformations. As noted earlier, 121 (11111) is the only number that can survive subtraction from 242 and emerge unchanged. But any palindrome pattern will find itself the same after left-right inversion.
More fascinating are the patterns that survive after the combination transform (combo). Among the 9 numbers that appear in each chart, 121, 225, and 65 each go through the combo process and eventually end up as their original numerical selves.
It would be interesting to investigate all 243 numbers, and see which of them share this quality.
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