Let n be a positive integer. A positive n-licht has two parts:
n is termed the dimensionality.
Later the restriction of positivity will be examined, but for now it eliminates a number of distracting complications without losing substance.
If A is an n-licht (positive or not), its divisors are represented by A_{𝗗} and its lengths by A_{𝗟}. The components of a generic n-licht A can be written between two shallow angle brackets, as the following:
A = ⟨ A_{𝗗}; A_{𝗟} ⟩
which can be expanded by enumerating the components:
A = ⟨ A_{𝗗0}, A_{𝗗1}, A_{𝗗2} … A_{𝗗n−1}; A_{𝗟0}, A_{𝗟1}, A_{𝗟2} … A_{𝗟n−1} ⟩
and which might be split between two lines for readability:
A = ⟨ | A_{𝗗0}, | A_{𝗗1}, | A_{𝗗2} | … | A_{𝗗n−1} | ; |
A_{𝗟0}, | A_{𝗟1}, | A_{𝗟2} | … | A_{𝗟n−1} | ⟩ |
The two n-tuples are not independent, the divisors indirectly determining the ratios of the lengths. The usual practice to create n-licht A is to arbitrarily select all the divisors, and then to specify A_{𝗟0}; this is enough to establish the remaining lengths. How to do this will be explained shortly.
As would be expected, A = B if and only if A_{𝗗} = B_{𝗗} and A_{𝗟} = B_{𝗟}.
The following table gives some typographical information about how this page was produced:
HTML source code | rendering |
---|---|
<i>A</i><sub>𝗗3</sub> | A_{𝗗3} |
<i>A</i><sub>𝗟<i>n</i></sub> | A_{𝗟n} |
<i>A</i><sub>𝗥<i>n</i>−1</sub> | A_{𝗥n−1} |
The motivation for the relationship between divisors and lengths is best expressed by way of the decrement operation. By definition:
dec (A) = ⟨ | A_{𝗗0}, | A_{𝗗1}, | A_{𝗗2}, | A_{𝗗3} | … | A_{𝗗n−1} | ; |
A_{𝗟n−1} ÷ A_{𝗗n−1}, | A_{𝗟0} ÷ A_{𝗗0}, | A_{𝗟1} ÷ A_{𝗗1}, | A_{𝗟2} ÷ A_{𝗗2} | … | A_{𝗟n−2} ÷ A_{𝗗n−2} | ⟩ |
In other words,
Here is a numerical example with six-place approximations. Given this:
A = ⟨ | 5, | 2, | 3, | 6 | ; |
1.200000, | 0.879082, | 1.609969, | 1.965687 | ⟩ |
then:
B = dec (A) = ⟨ | 5, | 2, | 3, | 6 | ; |
0.327615, | 0.240000, | 0.439541, | 0.536656 | ⟩ |
The following equality of ratios is preserved by the decrement operation:
If B = dec (A), then A_{𝗟0} : A_{𝗟1} : A_{𝗟2} : A_{𝗟3} = B_{𝗟0} : B_{𝗟1} : B_{𝗟2} : B_{𝗟3} If decrementing an n-licht does not satisfy this relation, then the n-licht is invalid. |
The relation of ratios can equivalently be written:
A_{𝗟0} : B_{𝗟0} = A_{𝗟1} : B_{𝗟1} = A_{𝗟2} : B_{𝗟2} = A_{𝗟3} : B_{𝗟3}
which happens to equal 3.662842 : 1 in this case.
Decrement has an inverse function called increment. By definition:
inc (A) = ⟨ | A_{𝗗0}, | A_{𝗗1}, | A_{𝗗2}, | … | A_{𝗗n−2} | A_{𝗗n−1} | ; |
A_{𝗟1} × A_{𝗗0}, | A_{𝗟2} × A_{𝗗1} | A_{𝗟3} × A_{𝗗2} | … | A_{𝗟n−1} × A_{𝗗n−2}, | A_{𝗟0} ÷ A_{𝗗n−1}, | ⟩ |
In other words,
As expected, inc (dec (A)) = dec (inc (A)).
Certain quantities, called the ratios, are helpful in performing calculations. They form an n-tuple notated with the subscript 𝗥, and are entirely dependent on the divisors. The ratios are not an "official" part of the n-licht.
For an n-licht A, define A_{𝗗gm} as the geometric mean of the divisors, in other words the nth root of their product. Then for each n, define A_{𝗥n} = A_{𝗗gm} ÷ A_{𝗗n}. Now, given all of its divisors and its first length, n-licht A can be conveniently produced:
A = ⟨ | A_{𝗗0}, | A_{𝗗1}, | A_{𝗗2}, | A_{𝗗3} | … | A_{𝗗n−1} | ; |
A_{𝗟0}, | A_{𝗟0} × A_{𝗥0}, | A_{𝗟1} × A_{𝗥1}, | A_{𝗟2} × A_{𝗥2}, | … | A_{𝗟n−2} × A_{𝗥n−2} | ⟩ |
If the n-licht is of high dimensionality, this formula might generate excessive cumulative error, and a different method might be preferred:
component | as above | alternate |
---|---|---|
A_{𝗟0} | A_{𝗟0} | A_{𝗟0} |
A_{𝗟1} | A_{𝗟0} × A_{𝗥0} | A_{𝗟0} × A_{𝗗gm} ÷ A_{𝗗0} |
A_{𝗟2} | A_{𝗟1} × A_{𝗥1} | A_{𝗟0} × (A_{𝗗gm})^{2} ÷ (A_{𝗗0} × A_{𝗗1}) |
A_{𝗟3} | A_{𝗟2} × A_{𝗥2} | A_{𝗟0} × (A_{𝗗gm})^{3} ÷ (A_{𝗗0} × A_{𝗗1} × A_{𝗗2}) |
A_{𝗟4} | A_{𝗟3} × A_{𝗥3} | A_{𝗟0} × (A_{𝗗gm})^{4} ÷ (A_{𝗗0} × A_{𝗗1} × A_{𝗗2} × A_{𝗗3}) |
and so forth |
Note that the A_{𝗗n} are integers, so a lengthy multiplicative series does not lead to error. Although A_{𝗗gm} is a real number, there are methods to find its higher powers less error-susceptible than plain repeated multiplication.
It might be said that an n-licht has n + 1 degrees of freedom: one for each divisor, and one for a length.
Some other important quantities:
Calculation of these roots can lead to a problem when a even-indexed root is required of a negative number — there will be no solution among the real numbers. That is a major reason to restrict n-licht components to positivity. An extension to complex numbers might be seen as a way to handle this, the divisors probably being either the Gaussian integers or the Eisenstein integers, but then arises the decision of which of multiple roots to select.
The unit function divides an n-licht by its magnitude, hence mag (unit (A)) = 1. Restricting n-lichts to positivity prevents the division-by-zero problem. Even if positivity is not required, divisors can never be zero.
There is a simple rotation operation; all divisors and all lengths are moved through the same distance. Here is an example, rotating to the right by one position:
rot (A, +1) = ⟨ | A_{𝗗n−1} | A_{𝗗0}, | A_{𝗗1}, | A_{𝗗2} | … | A_{𝗗n−2} | ; |
A_{𝗟n−1} | A_{𝗟0}, | A_{𝗟1}, | A_{𝗟2} | … | A_{𝗟n−2} | ⟩ |
For rotation to the left, the second operand will be a negative number. Ratios, if used, are rotated in the same manner.
If divisors are permuted into a sequence that is not a rotation, the nature of the relationship among the lengths becomes more complicated. Examples:
4-licht | magnitude | step |
---|---|---|
⟨ 1, 2, 3, 4; 1.000000, 2.213364, 2.449490, 1.807204 ⟩ | 1.769228 | 2.213364 |
⟨ 1, 2, 4, 3; 1.000000, 2.213364, 2.449490, 1.355403 ⟩ | 1.646453 | |
⟨ 1, 3, 2, 4; 1.000000, 2.213364, 1.632993, 1.807204 ⟩ | 1.598678 | |
⟨ 1, 3, 4, 2; 1.000000, 2.213364, 1.632993, 0.903602 ⟩ | 1.344323 | |
⟨ 1, 4, 2, 3; 1.000000, 2.213364, 1.224745, 1.355403 ⟩ | 1.384496 | |
⟨ 1, 4, 3, 2; 1.000000, 2.213364, 1.224745, 0.903602 ⟩ | 1.251033 |
Arithmetic can be performed on n-lichts. If A_{𝗗} = B_{𝗗}, then addition is:
A + B = ⟨ | A_{𝗗0}, | A_{𝗗1}, | A_{𝗗2} | … | A_{𝗗n−1} | ; |
A_{𝗟0} + B_{𝗟0}, | A_{𝗟1} + B_{𝗟1}, | A_{𝗟2} + B_{𝗟2} | … | A_{𝗟n−1} + B_{𝗟n−1} | ⟩ |
If A_{𝗗} ≠ B_{𝗗}, addition is not defined.
Addition is commutative and associative.
In an additive identity, all the lengths would be zero.
If nonpositive n-lichts are allowed, subtraction is defined in the obvious manner.
Multiplication by a scalar S is simple, and is commutative:
A × S = S × A = ⟨ | A_{𝗗0}, | A_{𝗗1}, | A_{𝗗2} | … | A_{𝗗n−1} | ; |
A_{𝗟0} × S, | A_{𝗟1} × S, | A_{𝗟2} × S | … | A_{𝗟n−1} × S | ⟩ |
Scalar multiplication distributes over addition:
A × S + B × S = (A + B) × S
Two n-lichts can be multiplied, even if their divisors are different, as long as they have the same dimensionality. By definition, all respective components are multiplied:
A × B = ⟨ | A_{𝗗0} × B_{𝗗0}, | A_{𝗗1} × B_{𝗗1}, | A_{𝗗2} × B_{𝗗2} | … | A_{𝗗n−1} × B_{𝗗n−1} | ; |
A_{𝗟0} × B_{𝗟0}, | A_{𝗟1} × B_{𝗟1}, | A_{𝗟2} × B_{𝗟2} | … | A_{𝗟n−1} × B_{𝗟n−1} | ⟩ |
This operation is commutative, associative, and it distributes over addition. In a multiplicative identity, all the divisors and lengths would equal one. Also:
vol (A × B) = | vol (A) × | vol (B) |
mag (A × B) = | mag (A) × | mag (B) |
step (A × B) = | step (A) × | step (B) |
If ratios are needed, their respective components are similarly multiplied.
If m ≠ n, then the product of an m-licht and n-licht is not defined.
The catenation operation takes an m-licht and an n-licht, where m need not equal n, and produces an (m + n)-licht. An ampersand & is the symbol of this rather complicated operation. Because it is difficult to provide a manageable formula, what follows is a procedure.
Let A be an m-licht, and B be an n-licht. Their divisors can be written out:
A_{𝗗} = (
A_{𝗗0},
A_{𝗗1},
A_{𝗗2} …
A_{𝗗m−1}
)
B_{𝗗} = (
B_{𝗗0},
B_{𝗗1},
B_{𝗗2} …
B_{𝗗n−1}
)
Then the ordered (m + n)-tuple of divisors of C = A & B is produced by ordinary catenation:
C_{𝗗} = ( A_{𝗗0}, A_{𝗗1}, A_{𝗗2} … A_{𝗗m−1}, B_{𝗗0}, B_{𝗗1}, B_{𝗗2} … B_{𝗗n−1} )
To be explicit:
With the divisors established, the ordered (m + n)-tuple of ratios for C can be formed. Because there is no simple way to adapt the ratios from A and B, it will probably be easiest to find C's ratios from scratch. Therefore, calculate C_{𝗗gm} as the geometric mean of all of C's divisors. Then C_{𝗥0} = C_{𝗗gm} ÷ C_{𝗗0}; C_{𝗥1} = C_{𝗗gm} ÷ C_{𝗗1}; et cetera.
Now make the temporary assumption that C_{𝗟0} equals 1, and find the lengths of C:
At this point, define (m + n)-licht E = unit (C) × mag (A) × mag (B). To put it another way, the magnitude of C is scaled to the desired value, with no change in divisors. Now A & B can by definition equal E. Clearly, mag (A) × mag (B) = mag (E).
Catenation is designed to be distributive over addition. Given m-lichts P and Q, with n-licht R, this means:
(P + Q) & R = (P & R) + (Q & R)
R & (P + Q) = (R & P) + (R & Q)
Catenation is associative: (P & Q) & R = P & (Q & R) for any combination of dimensionalities. Meanwhile, the identity for catenation is the 0-licht.
As before, let A be an m-licht, and B be an n-licht. Then the step of A & B is the weighted geometric mean of the steps of A and B, as follows:
(A & B)_{𝗗gm} = ((A_{𝗗gm})^{m} × (B_{𝗗gm})^{n})^{1÷(m+n)}