In the previous post below, I introduced what I call the gear group of rotations, which has a representation in the set of S|T units, which comprise the preons of the standard model, defined previously. The gear group is a mathematical group, expressed as two, reciprocal, rotations, which act just like mechanical gears. The requirement for a set of numbers to qualify as a mathematical group is that it meets a short list of criteria. Wikipedia offers this definition:
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. For example, the set of integers is a group under the operation of addition.
The group axioms are:
- Closure: The result of the binary operation with elements in the set, must also be a member of the set.
- Associativity: Changing the order of a sequence of binary operations must not make a difference in the result.
- Identity element: There must be an identity element in the set.
- Inverse element: Each element in the set must have an inverse element in the set.
The set of integers and the set of rational numbers are examples of mathematical groups. The Wikipedia article shows a proof that the set of integers satisfies the group axioms under the binary operation of addition:
A familiar group is the group of integers under addition. Let Z be the set of integers, {…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …}, and let the symbol “+” indicate the operation of addition. Then (Z,+) is a group.
Proof:
- Closure: If a and b are integers then a + b is an integer.
- Associativity: If a, b, and c are integers, then (a + b) + c = a + (b + c).
- Identity element: 0 is an integer and for any integer a, 0 + a = a + 0 = a.
- Inverse elements: If a is an integer, then the integer −a satisfies the inverse rules: a + (−a) = (−a) + a = 0.
This group is also abelian because a + b = b + a.
If we extend this example further by considering the integers with both addition and multiplication, which (sic) forms a more complicated algebraic structure called a ring. (But, note that the integers with multiplications are not a group)
Note that the integers do not form a group under the binary operation of multiplication. The reason for this is that the integers include zero, and thus there is no identity element under multiplication. However, in the RSM, the operationally interpreted (OI) rational numbers, which are equivalent to the set of integers, replace zero with 1/1, the identity element. Nevertheless, we have not defined the multiplication operation for OI numbers, because we couldn’t see a need for such an operation. Given SUDRs and TUDRs, we combine them into SUDR|TUDR combinations by addition. For example, one SUDR combined with one TUDR (its inverse), results in the identity element, under addition,
ds|dt = 1|2 + 2|1 = 3|3 = 1|1,
but multiplying one SUDR by one TUDR,
ds|dt = 1|2 * 2|1 = 2|2 = 1|1,
while it also results in the identity element, didn’t make any sense, from the perspective of the definition of the binary operation; that is, in scalar mathematics, multiplication simply is a short hand for stating how many instances of a number are to be added together, like stating 1 of x, or 10 of y, or 3 of z, but the idea of -1 of x, or -10 of y, or -3 of z, is just nonsensical.
Fortunately, however, the so-called non-zero, quantitatively interpreted (QI), rational numbers do form a group under multiplication, according to Wikipedia. So, it appears that, in order to introduce the multiplication operation into the relations of S|T units, as a representation of a group under multiplication, we need to revert to the ordinary QI rational number. At first glance, it appeared to me that this would be counterproductive, given the central role of the OI rational number in our theoretical development to this point, and so I balked at the idea, initially.
However, I soon realized that when we consider the S|T units as a representation of the gear group of rotations, we are not combining SUDRs and TUDRs individually, but only combinations of SUDRs and TUDRs, as S|T units. Since these combination units have both positive and negative elements (SUDRs & TUDRs), simultaneously, we can add additional SUDRs, and/or additional TUDRs, to existing S|T units, and we can add multiple S|T units together, because they are all elements, or sums of elements, which are included in the representation of the group of OI rational numbers (integers), under addition. Consequently, the set of summed SUDRs and TUDRs, as S|T units, represent a subgroup within the group of SUDRs and TUDRs, under addition.
But if we want to change the binary operation of this subgroup to multiplication, then the set of S|T units must qualify as a representation of the group of QI rational numbers (non-zero rationals), under multiplication, in its own right. In other words, if the gear group of rotations actually satisfies the four axioms of a group, under the multiplication operation, which we’ve yet to show, then the S|T unit representation of this group opens up an entirely new range of possibilities for describing relations between S|T units, and combinations of S|T units, such as preons.
Recall that the identity element of the gear group is the unit gear ratio, 1:1, as shown in figure 1 of the previous post. Clearly, increasing the unit gear ratio from 1:1, to 2:1, or decreasing it, to 1:2, is equivalent to adding a unit positive, or unit negative, number, respectively. To do this, we double the size of one, or the other, of the two gears. For example, we can double the SUDR gear, as shown in figure 1 below. Doubling the SUDR displacement in the S|T unit, from ds|dt = 1|2 = -1, to ds|dt = 2|4 = -2, is equivalent to doubling the diameter of the SUDR gear.
Figure 1. Unit Gear Ratio Reduced from 1:1 to 1:2
In the unit gear ratio, illustrated in the previous post below, one revolution of the SUDR gear is equal to one revolution of the TUDR gear, but decreasing the gear ratio from 1:1, to 1:2, by doubling the size of the SUDR gear, as shown in figure 1 above, means that for every revolution of the SUDR gear, now the TUDR gear must revolve two times. If we use the space and time units of the SUDR and TUDR, as a representation of the elements of the integer group, to calculate the net result of this relationship, we get (ignoring the inward units for simplicity)
ds|dt = 2|4 + 2|1 = 4|5 = -1,
which indicates that the sum at the apex is unbalanced by one displacement unit of progression in the SUDR (or negative) “direction.” In other words, it indicates that the SUDR “gear” is twice as big as the “TUDR” “gear,” at the apex connection. But this is in terms of the ratio of the number of individual space units and time units of the OI rational numbers (integer) group, which represent net space|time displacements. If we express this same relationship in terms of the ratio of the number of SUDR units (2) and and the number of TUDR units (1), instead of in terms of the space and time units that make them up, we get a different number altogether, even though they are both expressing the same value. The new number is a ratio of ratios.
For example, if we consider the value at the A apex of a triplet, the S|T units are reciprocal ratios of SUDR units and TUDR units, which are units of unit negative and unit positive space|time displacements, not reciprocal ratios of space units and time units. Yet, the two expressions are equivalent. Hence, we can write the value of the A apex with two SUDRs connected to one TUDR as,
S|T = 2|1 = [((1|2) + (1|2)) + (2|1)] = (2|4 + 2|1) = 4|5 = -1.
So, now that we see that we have equivalent, but different numbers, representing the same magnitude, the question is, are the properties of the two different numbers the same? In particular, we want to know, if we have two numbers, representing S|T units, can we multiply them together, and will the result be another S|T unit that is also an element in the set, and will there be an identity element and inverse element, and so forth, forming a representation of a group under multiplication?
Of course, the reason this question is so important is because we must multiply two gear ratios, to get a new gear ratio. We only add them in determining the relative sizes of the gears. So, we need to know if the results of the multiplication operations with S|T numbers qualify them as a representation of the group of reciprocal rotations we call the gear group.
I think that the answer to this question is yes, but I can’t demonstrate it completely, yet. However, I can say some things about the differences between the two kinds of numbers, though. Notice that the sign of the OI polarity, or OI “direction,” in the S|T unit is reversed, relative to the “direction” of the space|time ratio; that is, while more SUDRs than TUDRs is an imbalance in the negative “direction,” the greater number is on the left of the symbol, “|”, used to denote the reciprocal relation in OI numbers. This never happens in the space|time ratios, where the smaller number is always on the left, or on what we designate as the negative side of unity. Therefore, we will not use the same symbol in forming both kinds of numbers from now on. Instead, we will use the customary colon symbol, “:”, to denote the reciprocal relation of S:T numbers, instead. On this basis, we can see the equivalency of the two kinds of numbers, as follows:
- S:T = 1:1 = ds|dt = (1|2) + (2|1) = 3|3 = 0, and
- S:T = 2:1 = ds|dt = 2(1|2) + (2|1) = [(1|2)+(1|2)] + (2|1) = 4|5 = -1, and
- S:T = 1:2 = ds|dt = (1|2) + 2(2|1) = (1|2)+[(2|1)+(2|1)] = 5|4 = +1.
However, multiplying two S:T numbers results in a higher, or lower, ratio, which is not the same as a higher displacement, just as a higher, or lower, dimension, obtained through multiplication, is not the same as a higher quantity, obtained through multiplication. So the multiplication operation, while consistent in terms of the appropriate group, is not consistent across groups. What this amounts to is the necessity of switching from the OI reciprocal number (RN) (integers in the RSM), to the QI RN (rationals in the RSM), in the S:T representation of the gear group. This is an unexpected, but welcome, surprise, because I didn’t even know that there were rationals in the RSM!
However, just as the ordinary, non-RSM based, integers are different in some important ways from the ordinary, non-zero, rationals, the QI RNs are different in some important ways from the OI RNs. For example,
S:T = 2:1 * 2:1 = 4:1 = -3 ≠ (-1) * (-1) = 1,
even though,
S:T = 4:1 = [((1|2)+(1|2)+(1|2)+(1|2)) + (2|1)] = 4|8 + 2|1 = 6|9 = -3.
That is to say, in the multiplication operation of the elements of the S:T group, 2:1 ≠ 2|1, even if the reverse “direction” of the polarity of the S:T number is noted. The reason for this is that the necessity of using the QI RN, in lieu of the OI RN, to accommodate multiplication, changes the value of 2:1 to the QI value of “.5”, from the OI value of “-1”. Hence, with QI RNs,
S:T = 2:1 * 2:1 = 4:1 = (.5)*(.5) = .25,
not -3, as it would with OI RNs. Yet, at the same time, if we have a 1:2 SUDR unit, multiplied by a 2:1 TUDR unit, we also get the identity element:
S:T = 1:2 * 2:1 = 2:2 = (.5) * 2 = 1 = 1:1,
and the rest of the properties of the group axioms, under multiplication, are likewise satisfied, I believe.
It turns out then, that I think that we have discovered something new and startling: The OI RN is the reciprocal of the QI RN; that is, as Hestenes points out, following Grassmann and Clifford, it is true that there are two interpretations of number possible, but what is more, we have found that the quantitative and operational interpretations of number that they identified, are actually only two, reciprocal, aspects of the same thing. The truth is that “half of much of something” is the reciprocal of “twice as much of something,” and any RN may be interpreted either way, depending upon the desired outcome.
This is a profound and fundamental concept, and I will treat it in more detail on The New Math blog, as soon as I can get to it. In the meantime, I think we have found the way to multiply our RNs. We simply need to use the “half as much” interpretation, rather than the “twice as much” interpretation, when we multiply them.