Generalized Lichtenberg ratios.
Version of Thursday 26 December 2019.
Dave Barber's other pages.

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 A𝗗 and its lengths by A𝗟. The components of a generic n-licht A can be written between two shallow angle brackets, as the following:

A = ⟨ A𝗗; A𝗟

which can be expanded by enumerating the components:

A = ⟨ A𝗗0, A𝗗1, A𝗗2A𝗗n−1; A𝗟0, A𝗟1, A𝗟2A𝗟n−1

and which might be split between two lines for readability:

A = ⟨ A𝗗0, A𝗗1, A𝗗2 A𝗗n−1 ;
A𝗟0, A𝗟1, A𝗟2 A𝗟n−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 A𝗟0; 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 A𝗗 = B𝗗 and A𝗟 = B𝗟.

The following table gives some typographical information about how this page was produced:

HTML source coderendering
<i>A</i><sub>&#x1D5D7;3</sub> A𝗗3
<i>A</i><sub>&#x1D5DF;<i>n</i></sub> A𝗟n
<i>A</i><sub>&#x1D5E5;<i>n</i>&minus;1</sub> A𝗥n−1

The motivation for the relationship between divisors and lengths is best expressed by way of the decrement operation. By definition:

dec (A) = ⟨ A𝗗0, A𝗗1, A𝗗2, A𝗗3 A𝗗n−1 ;
A𝗟n−1 ÷ A𝗗n−1, A𝗟0 ÷ A𝗗0, A𝗟1 ÷ A𝗗1, A𝗟2 ÷ A𝗗2 A𝗟n−2 ÷ A𝗗n−2

In other words,

Here is a numerical example with six-place approximations. Given this:

A = ⟨5,2,3,6;


B = dec (A) = ⟨5,2,3,6;

The following equality of ratios is preserved by the decrement operation:

If B = dec (A), then A𝗟0 : A𝗟1 : A𝗟2 : A𝗟3 = B𝗟0 : B𝗟1 : B𝗟2 : B𝗟3

If decrementing an n-licht does not satisfy this relation, then the n-licht is invalid.

The relation of ratios can equivalently be written:

A𝗟0 : B𝗟0 = A𝗟1 : B𝗟1 = A𝗟2 : B𝗟2 = A𝗟3 : B𝗟3

which happens to equal 3.662842 : 1 in this case.

Decrement has an inverse function called increment. By definition:

inc (A) = ⟨ A𝗗0, A𝗗1, A𝗗2, A𝗗n−2 A𝗗n−1 ;
A𝗟1 × A𝗗0, A𝗟2 × A𝗗1 A𝗟3 × A𝗗2 A𝗟n−1 × A𝗗n−2, A𝗟0 ÷ A𝗗n−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 A𝗗gm as the geometric mean of the divisors, in other words the nth root of their product. Then for each n, define A𝗥n = A𝗗gm ÷ A𝗗n. Now, given all of its divisors and its first length, n-licht A can be conveniently produced:

A = ⟨ A𝗗0, A𝗗1, A𝗗2, A𝗗3 A𝗗n−1 ;
A𝗟0, A𝗟0 × A𝗥0, A𝗟1 × A𝗥1, A𝗟2 × A𝗥2, A𝗟n−2 × A𝗥n−2

If the n-licht is of high dimensionality, this formula might generate excessive cumulative error, and a different method might be preferred:

componentas abovealternate
A𝗟1 A𝗟0 × A𝗥0 A𝗟0 × A𝗗gm ÷ A𝗗0
A𝗟2 A𝗟1 × A𝗥1 A𝗟0 × (A𝗗gm)2 ÷ (A𝗗0 × A𝗗1)
A𝗟3 A𝗟2 × A𝗥2 A𝗟0 × (A𝗗gm)3 ÷ (A𝗗0 × A𝗗1 × A𝗗2)
A𝗟4 A𝗟3 × A𝗥3 A𝗟0 × (A𝗗gm)4 ÷ (A𝗗0 × A𝗗1 × A𝗗2 × A𝗗3)
and so forth

Note that the A𝗗n are integers, so a lengthy multiplicative series does not lead to error. Although A𝗗gm 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) = ⟨ A𝗗n−1 A𝗗0, A𝗗1, A𝗗2 A𝗗n−2 ;
A𝗟n−1 A𝗟0, A𝗟1, A𝗟2 A𝗟n−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:

⟨ 1, 2, 3, 4; 1.000000, 2.213364, 2.449490, 1.807204 ⟩ 1.7692282.213364
⟨ 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 A𝗗 = B𝗗, then addition is:

A + B = ⟨ A𝗗0, A𝗗1, A𝗗2 A𝗗n−1 ;
A𝗟0 + B𝗟0, A𝗟1 + B𝗟1, A𝗟2 + B𝗟2 A𝗟n−1 + B𝗟n−1

If A𝗗B𝗗, 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 = ⟨ A𝗗0, A𝗗1, A𝗗2 A𝗗n−1 ;
A𝗟0 × S, A𝗟1 × S, A𝗟2 × S A𝗟n−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 = ⟨ A𝗗0 × B𝗗0, A𝗗1 × B𝗗1, A𝗗2 × B𝗗2 A𝗗n−1 × B𝗗n−1 ;
A𝗟0 × B𝗟0, A𝗟1 × B𝗟1, A𝗟2 × B𝗟2 A𝗟n−1 × B𝗟n−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 mn, 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:

A𝗗 = ( A𝗗0, A𝗗1, A𝗗2A𝗗m−1 )
B𝗗 = ( B𝗗0, B𝗗1, B𝗗2B𝗗n−1 )

Then the ordered (m + n)-tuple of divisors of C = A & B is produced by ordinary catenation:

C𝗗 = ( A𝗗0, A𝗗1, A𝗗2A𝗗m−1, B𝗗0, B𝗗1, B𝗗2B𝗗n−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 C𝗗gm as the geometric mean of all of C's divisors. Then C𝗥0 = C𝗗gm ÷ C𝗗0; C𝗥1 = C𝗗gm ÷ C𝗗1; et cetera.

Now make the temporary assumption that C𝗟0 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 = ((A𝗗gm)m × (B𝗗gm)n)1÷(m+n)