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Lichtenberg ratios for real numbers.

§R1A Introduction. Let b be a positive integer. A b-licht contains two parts:

• an ordered b-tuple of positive real numbers, called the divisors;
• an ordered b-tuple of positive real numbers, called the lengths.

b is termed the breadth; the smallest nontrivial value is two. Some researchers might admit the degenerate breadth of zero, although that is not done in this report. When the breadth need not be specified, the b- prefix can be omitted.

When lichts of different breadths are discussed within one context, they might be called a-licht, b-licht, and c-licht. An alternate notation for this is b₁-licht, b₂-licht, and b₃-licht.

Some notations for parts of b-licht A:

• its breadth, namely b, is also written A_𝗕;
• the b-tuple of its divisors is A_𝗗;
• the b-tuple of its lengths is A_𝗟.

A ≡ ⟨ A_𝗗; A_𝗟 ⟩

which can be expanded by enumerating the components:

A ≡ ⟨ A_𝗗0, A_𝗗1, A_𝗗2 … A_𝗗b−1; A_𝗟0, A_𝗟1, A_𝗟2 … A_𝗟b−1 ⟩

and which is often split between two lines for readability:

A ≡ ⟨	A𝗗0,	A𝗗1,	A𝗗2	…	A𝗗b−1	;
	A𝗟0,	A𝗟1,	A𝗟2	…	A𝗟b−1	⟩

The two b-tuples are not independent, the divisors indirectly determining the ratios of the lengths. The usual practice to create 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 in §R2B below.

§R1B Preliminary comments.

A = B if and only if A_𝗕 = B_𝗕, A_𝗗 = B_𝗗, and A_𝗟 = B_𝗟. Inequalities such as A < B are meaningful only if A_𝗕 = B_𝗕 and A_𝗗 = B_𝗗, in which case A < B is logically equivalent to A_𝗟0 < B_𝗟0.

If all the divisors are equal, all the lengths will consequently be equal, and the licht is termed flat.

It can be said that a b-licht has b + 1 degrees of freedom: one for each divisor, and one for a length.

This report presumes that the b of a b-licht is finite. However, some researchers may want to produce a one-way infinite licht by allowing b to range among all the positive or nonnegative integers; or a two-way infinite licht by allowing b to range among all integers.

For multiplication, this report generally uses an explicit times symbol × rather than juxtaposition or a centered dot; and for division, the traditional obelus ÷.

It is difficult to find useful examples of lichts that employ only integers, as roots are frequently extracted and then used in subsequent calculations. Hence most examples here will display real numbers to six-place accuracy. Many integer lichts that are limited to small numbers are conspicuously trivial or contrived, for example:

⟨	2, 2, 4, 1, 2, 2	;
	2, 2, 2, 1, 2, 2	⟩

There does exist a method to generate a large class of nontrivial integer lichts that will typically contain large numbers.

§R1C About positivity.

Later (§C1A) the restriction of positivity will be lifted, but for now it eliminates a myriad of distracting complications without losing substance. (The informal term is "can of worms".) If lengths can be positive or negative, then zero will show up eventually. Moreover, a thorough treatment of allowing lengths to be negative entails the introduction of complex numbers, which have multiple roots that must be managed.

If any length of a licht were allowed to be zero, all lengths would be zero; such is called a zero licht. Here is an open question: if two zero lichts of the same breadth have different divisors, is it a good idea to describe them as equal?

Whatever numbers are used, divisors can never be zero.

§R2A Some useful quantities. Of b-licht A:

• The volume (A_𝗩) is the product of the lengths.
• The magnitude (A_𝗠) is the bth root of the volume.
• The step (A_𝗦) is the bth root of the product of the divisors.

The terms length and volume are suggested by interpreting the lengths as the edges of a b-dimensional rectangular solid, as exemplified by the 2-dimensional case in §R3E below.

Certain quantities, called the ratios, are helpful in performing calculations. They are a b-tuple of positive real numbers notated with the subscript 𝗥, and are entirely dependent on the divisors. The ratios are not an "official" part of a b-licht. Given a b-licht A, define:

A_𝗥i = A_𝗦 ÷ A_𝗗i
for each integer i: 0 ≤ i < b

This formula is why divisors are called divisors.

§R2B Usual method to create a licht from scratch. If A_𝗗 and A_𝗟0 have been supplied, possibly by arbitrary choice, b-licht A can be produced using the ratios from above:

A = ⟨	A𝗗0,	A𝗗1,	A𝗗2,	A𝗗3	…	A𝗗b−1	;
	A𝗟0,	A𝗟0 × A𝗥0,	A𝗟1 × A𝗥1,	A𝗟2 × A𝗥2,	…	A𝗟b−2 × A𝗥b−2	⟩

If the licht is of large breadth, and if non-exact arithmetic is being used, the researcher should be alert for excessive cumulative error.

Although it is not used, the next term in the sequence of lengths would have been A_𝗟b−1 × A_𝗥b−1. Noteworthily, it equals A_𝗟0.

§R2C Unitization.

• The unit_len operation divides each length by the magnitude. Divisors are unaffected. This is frequently useful.
• The unit_div operation divides each divisor by the step. Lengths are unaffected. This is rarely, but occasionally, useful.

Numerical example:

A = ⟨	0.318005,	3.299363,	0.704850,	1.474858	;
	1.403548,	4.510462,	1.397072,	2.025581	⟩
	mag = 2.057331		step = 1.021946
unit_len (A) = ⟨	0.318005,	3.299363,	0.704850,	1.474858	;
	0.682218,	2.192385,	0.679070,	0.984568	⟩
	mag = 1.000000		step = 1.021946
unit_div (A) = ⟨	0.311176,	3.228512,	0.689714,	1.443187	;
	1.403548,	4.510462,	1.397072,	2.025581	⟩
	mag = 2.057331		step = 1.000000

§R2D Alternate method to create a licht from scratch. If A_𝗗 and the desired value of A_𝗠 have been supplied:

• Create a temporary licht B by the method of §R2B, using A_𝗗 and a value of 1 for B_𝗟0.
• Let licht C = unit_len (B).
• Multiply each length in C by the desired value of A_𝗠, producing the final result.

This differs from the method of §R2B only in the scaling of the lengths.

§R3 Manipulators. Four important operations are called manipulators: decrement, increment, cycle, and reverse. They do not change the breadth. The first two of these are scaling operations, particularly significant as they illustrate the rationale for lichts.

§R3A Decrement. The motivation for the relationship between divisors and lengths is best expressed through this operation. By definition:

dec (A) = ⟨	A𝗗0,	A𝗗1,	A𝗗2,	A𝗗3	…	A𝗗b−1	;
	A𝗟b−1 ÷ A𝗗b−1,	A𝗟0 ÷ A𝗗0,	A𝗟1 ÷ A𝗗1,	A𝗟2 ÷ A𝗗2	…	A𝗟b−2 ÷ A𝗗b−2	⟩

In other words,

• Each A_𝗟i is divided by its respective A_𝗗i.
• Then all the A_𝗟i are cycled one position to the right, with the last moving to the first position.
• A_𝗗 is unaffected.

Here is a numerical example. Given this:

A = ⟨	0.412548,	4.280254,	0.914400,	1.913330	;
	2.179367,	7.003641,	2.169311,	3.145233	⟩

then:

dec (A) = ⟨	0.412548,	4.280254,	0.914400,	1.913330	;
	1.643853,	5.282705,	1.636267,	2.372386	⟩

There is an equality of ratios that preserved by the decrement operation. If B = dec (A), then: A𝗟0 : A𝗟1 : A𝗟2 … = B𝗟0 : B𝗟1 : B𝗟2 … This behavior is foundational to lichts. If decrementing a licht does not satisfy this relation, then the licht was defective to begin with.

The relation of ratios can equivalently be written thus:

A_𝗟0 : B_𝗟0 = A_𝗟1 : B_𝗟1 = A_𝗟2 : B_𝗟2 …

which happens to equal 1.325768 : 1 in this case. The step of both A and dec (A) is 1.325768.

§R3B Increment is the inverse operation of decrement. By definition:

inc (A) = ⟨	A𝗗0,	A𝗗1,	A𝗗2,	…	A𝗗b−2	A𝗗b−1	;
	A𝗟1 × A𝗗0,	A𝗟2 × A𝗗1	A𝗟3 × A𝗗2	…	A𝗟b−1 × A𝗗b−2,	A𝗟0 × A𝗗b−1,	⟩

In other words,

• All the A_𝗟i are cycled one position to the left, with the first moving to the last position.
• Then each A_𝗟i is multiplied by what is now its respective A_𝗗i.
• A_𝗗 is unaffected.

Of course, inc (dec (A)) = A = dec (inc (A)).

§R3C Cycle is a very simple operation: all divisors and all lengths are moved through the same distance. Here is an example, cycling to the right by one position:

cyc (A, +1) = ⟨	A𝗗b−1	A𝗗0,	A𝗗1,	A𝗗2	…	A𝗗b−2	;
	A𝗟b−1	A𝗟0,	A𝗟1,	A𝗟2	…	A𝗟b−2	⟩

For cycling to the left, the second operand will be a negative number. Of course, cyc (A, i + j) = cyc (cyc (A, i), j) = cyc (cyc (A, j), i).

If the divisors are permuted into a sequence that is not a cycle, the nature of the relationship among the lengths becomes far more complicated.

Mentioned in §R2B is that A_𝗟b−1 × A_𝗥b−1= A_𝗟0. This gives some insight into why the cycle operation succeeds.

§R3D Reverse is unsurprising. If licht B = rev (A), the divisors of B are the same as the divisors of A, but in reverse sequence. Meanwhile, B_𝗠 is set to equal A_𝗠. To construct B, use the method of §R2D with B_𝗗 and A_𝗠.

For example, if:

A = ⟨	0.326600,	3.388535,	0.723900,	1.514719,	2.826322	;
	1.179875,	4.622447,	1.745469,	3.085216,	2.606187	⟩

it turns out that:

B = rev (A) = ⟨	2.826322,	1.514719,	0.723900,	3.388535,	0.326600	;
	4.805388,	2.175500,	1.837718,	3.248272,	1.226571	⟩

Either way, the volume is 76.544056, the magnitude 2.381126, and the step 1.279535. There is no obvious relationship between the lengths of A and those of rev (A).

As expected, rev (rev (A)), which is equivalently written rev² (A), equals A.

§R3E ISO Paper sizes. These are based on a Lichtenberg ratio, and this entire report is a broad generalization of that. Some of the ISO paper sizes are given in the table below, the figures rounded, in lineal or square millimeters.

size		divisors			lengths			area
A0	⟨	2,	1	;	1189,	841	⟩	1,000,000
A1 = dec (A0)	⟨	2,	1	;	841,	595	⟩	500,000
A2 = dec (A1)	⟨	2,	1	;	595,	420	⟩	250,000
A3 = dec (A2)	⟨	2,	1	;	420,	297	⟩	125,000
A4 = dec (A3)	⟨	2,	1	;	297,	210	⟩	62,500
A5 = dec (A4)	⟨	2,	1	;	210,	149	⟩	31,250
A6 = dec (A5)	⟨	2,	1	;	149,	105	⟩	15,625
size		divisors			lengths			area
B0	⟨	2,	1	;	1414,	1000	⟩	1,414,214
B1 = dec (B0)	⟨	2,	1	;	1000,	707	⟩	707,107
B2 = dec (B1)	⟨	2,	1	;	707,	500	⟩	353,553
B3 = dec (B2)	⟨	2,	1	;	500,	354	⟩	176,777
B4 = dec (B3)	⟨	2,	1	;	354,	250	⟩	88,388
B5 = dec (B4)	⟨	2,	1	;	250,	177	⟩	44,194
B6 = dec (B5)	⟨	2,	1	;	177,	125	⟩	22,097
B0 = inc (A0, 0.5)

Note that the official ISO standard truncates rather than rounds, sometimes giving numbers slightly less than these. However, this difference has little practical consequence, as the dimensions of a sheet of ordinary cellulose paper can vary with the humidity ("hygroexpansivity").

§R3F Comments.

Identities:

• cyc (dec (A), i) = dec (cyc (A, i))
• dec (rev (A)) = rev (dec (A)
• rev (cyc (A, +i) = cyc (rev (A), −i)

For any property that decrement has, increment will have a corresponding property.

As an avenue toward generalization, consider that when a licht is semi-decremented, all its lengths are divided by the square root of the step. Other fractional decrements are certainly possible, as decrement can be implemented as a scaling operation. A suggested notation is dec (A, 2) = dec (dec (A)); and more generally, dec (A, q) = dec (dec (A, q − 1)). Because of the inverse relationship, dec (A, −1) = inc (A). This can be readily adapted to fractional decrements.

§R4 Arithmetic can be performed on lichts, as follows.

§R4A Addition of b-lichts A and B is defined when A_𝗗 = B_𝗗:

A + B = ⟨	A𝗗0,	A𝗗1,	A𝗗2	…	A𝗗b−1	;
	A𝗟0 + B𝗟0,	A𝗟1 + B𝗟1,	A𝗟2 + B𝗟2	…	A𝗟b−1 + B𝗟b−1	⟩

This is concisely written A + B = ⟨ A_𝗗; A_𝗟 + B_𝗟 ⟩.

If A_𝗗 ≠ B_𝗗, addition is not defined. Addition is commutative and associative. As long as lengths are restricted from being zero, an additive identity cannot exist.

Identities:

• cyc (A, i) + cyc (B, i) = cyc ((A + B), i)
• dec (A) + dec (B) = dec (A + B)
• rev (A) + rev (B) = rev (A + B)

Also, A_𝗠 + B_𝗠 = (A + B)_𝗠.

§R4B Multiplication of lichts A and B is possible if A_𝗕 = B_𝗕 even when A_𝗗 ≠ B_𝗗. All respective components are multiplied:

A × B = ⟨	A𝗗0 × B𝗗0,	A𝗗1 × B𝗗1,	A𝗗2 × B𝗗2	…	A𝗗b−1 × B𝗗b−1	;
	A𝗟0 × B𝗟0,	A𝗟1 × B𝗟1,	A𝗟2 × B𝗟2	…	A𝗟b−1 × B𝗟b−1	⟩

Concisely: A × B = ⟨ A_𝗗 × B_𝗗; A_𝗟 × B_𝗟 ⟩. Similarly, C_𝗥 = A_𝗥 × B_𝗥.

This operation is commutative and associative; and it distributes over addition. In a multiplicative identity, all the divisors and lengths would equal one. Also:

• (A × B)_𝗩 = (A)_𝗩 × (B)_𝗩
• (A × B)_𝗠 = (A)_𝗠 × (B)_𝗠
• (A × B)_𝗦 = A_𝗦 × B_𝗦

Identities:

• cyc (A, i) × cyc (B, i) = cyc ((A × B), i)
• dec (A) × dec (B) = dec (A × B)
• rev (A) × rev (B) = rev (A × B)

§R4C Powers and roots. To raise a licht to a real-number power p, raise every component to that power:

A p = ⟨	(A𝗗0) p,	(A𝗗1) p,	(A𝗗2) p	…	(A𝗗b−1) p	;
	(A𝗟0) p,	(A𝗟1) p,	(A𝗟2) p	…	(A𝗟b−1) p	⟩

Similarly, to find the rth root of A, take the rth root of every component.

§R4D Division of lichts A and B is analogous to multiplication. All respective components are divided:

A ÷ B = ⟨	A𝗗0 ÷ B𝗗0,	A𝗗1 ÷ B𝗗1,	A𝗗2 ÷ B𝗗2	…	A𝗗b−1 ÷ B𝗗b−1	;
	A𝗟0 ÷ B𝗟0,	A𝗟1 ÷ B𝗟1,	A𝗟2 ÷ B𝗟2	…	A𝗟b−1 ÷ B𝗟b−1	⟩

If ratios are needed, their respective components are similarly divided.

The multiplicative inverse is:

B −1 = 1 ÷ B = ⟨	1 ÷ B𝗗0,	1 ÷ B𝗗1,	1 ÷ B𝗗2	…	1 ÷ B𝗗b−1	;
	1 ÷ B𝗟0,	1 ÷ B𝗟1,	1 ÷ B𝗟2	…	1 ÷ B𝗟b−1	⟩

§R4E Scalar multiplication and division are unsurprising. With licht A and real number S:

S × A = A × S = ⟨	A𝗗0,	A𝗗1,	A𝗗2	…	A𝗗b−1	;
	A𝗟0 × S,	A𝗟1 × S,	A𝗟2 × S	…	A𝗟b−1 × S	⟩
A ÷ S = ⟨	A𝗗0,	A𝗗1,	A𝗗2	…	A𝗗b−1	;
	A𝗟0 ÷ S,	A𝗟1 ÷ S,	A𝗟2 ÷ S	…	A𝗟b−1 ÷ S	⟩

Scalar operations on the lengths, as shown here, are the default because they are far more useful than scalar operations on the divisors, which nonetheless are perfectly valid. Note this independence:

• Multiplying all the lengths by a nonzero constant does not necessitate a change in the divisors.
• Multiplying all the divisors by a nonzero constant does not necessitate a change in the lengths.

§R4F Comments.

If several lichts can be added, their arithmetic mean can be found by dividing each length within their sum by the quantity of lichts.

If several lichts can be multiplied, their geometric mean can be found by taking the qth root of each divisor and length in their product, where q is the quantity of lichts.

§R5 Catenation. This operation takes a-licht A and b-licht B, where a need not equal b, and produces (a + b)-licht C. An ampersand & is the symbol of this rather complicated operation. Because it is difficult to provide a manageable formula, what follows is a procedure.

The divisors of A and B can be written out:

A_𝗗 = ( A_𝗗0, A_𝗗1, A_𝗗2 … A_𝗗a−1 )
B_𝗗 = ( B_𝗗0, B_𝗗1, B_𝗗2 … B_𝗗b−1 )

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

C_𝗗 = ( A_𝗗0, A_𝗗1, A_𝗗2 … A_𝗗a−1, B_𝗗0, B_𝗗1, B_𝗗2 … B_𝗗b−1 )

With the divisors established, the ordered (a + b)-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, in two steps:

• Set C_𝗥i = C_𝗦 ÷ C_𝗗i for each integer i: 0 ≤ i < a + b. (This is like the ratio calculation of §R2A.)
• Use the method of §R2D with C_𝗗 for the divisors, and A_𝗠 × B_𝗠 for the desired magnitude.

Catenation is associative: (X & Y) & Z = X & (Y & Z) for any three lichts. Meanwhile, the identity for catenation would be the 0-licht — if the researcher chooses to recognize such a licht.

Consider:

• addable a-lichts P and Q;
• addable b-lichts R and S.

Catenation is distributive over addition (but not multiplication). This means:

(P + Q) & R = (P & R) + (Q & R)

R & (P + Q) = (R & P) + (R & Q)

Contrast the following distributive expansions:

TRUE:	(P + Q) & (R + S) = (P & R) + (P & S) + (Q & R) + (Q & S)
FALSE:	(P & R) + (Q & S) = (P + Q) & (P + S) & (R + Q) & (R + S)

The false one is often undefined, but it still fails even if all four lichts have the same breadth and the same divisors, making everything addable.

As before, let A be an a-licht, and B be a b-licht. The reverse operation succeeds: rev (A) & rev (B) = rev (B & A). It is difficult to find useful relations for the other manipulators.

The step of A & B is the weighted geometric mean of the steps of A and B, as follows:

(A & B)_𝗦 = ((A_𝗦)^a × (B_𝗦)^b)^x
where x = 1 ÷ (a + b)

§R6 Weft. The input to this operation is quantity q of b-lichts, and the output is a (q × b)-licht. Weft resembles catenation in that the divisors of the inputs are combined, in a selected permutation, to produce the divisors of the output. Here is an example of the procedure, from which the general scheme can be surmised:

A, B, and C are 4-lichts. Their result, notated weft (A, B, C), will be a 12-licht. Call this result E. Its divisors, which form a 12-tuple of real numbers, are here displayed across four lines of text to make the pattern clear:

E𝗗 = (	A𝗗0, B𝗗0, C𝗗0,
	A𝗗1, B𝗗1, C𝗗1,
	A𝗗2, B𝗗2, C𝗗2,
	A𝗗3, B𝗗3, C𝗗3 )

The next steps resemble those of catenation. With the divisors established, the ordered 12-tuple of ratios for E can be formed, in two steps:

• Set E_𝗥i = E_𝗦 ÷ E_𝗗i for each integer i: 0 ≤ i < 12. (This is like the ratio calculation of §R2A.)
• Use the method of §R2D with E_𝗗 for the divisors, and A_𝗠 × B_𝗠 × C_𝗠 for the desired magnitude.

Identity:

rev (weft (A, B, C)) = weft (rev (C), rev (B), rev (A))

In contrast to catenation, weft distributes over multiplication (but not addition). For example, with six suitably defined lichts:

weft (A′ × A″, B′ × B″, C′ × C″) = weft (A′, B′, C′) × weft (A″, B″, C″))