Potency

Title: Potency
Subtitle: Informational Efficacy of Symbolisation to Exponentiated Bases

Introduction: Positional Notation
In positional numerical notation, we are used to as many different and distinguishable symbols in a set or inventory of characters as the cardinal number of the base to the first power being available for each place or position representing any power of that base. In this case, the number of items, say \(N\), that may be referred to as numbers, names, or words by a set of symbols, say \(S\), is the number \(\# S\) of symbols or characters in the set \(S\) equal to the base \(B\) raised to the power of the number \(n\) of times that different symbols \({}^{S}X\) appear in the name or word \({}^{N}X_{i}\) that represents or is used to call something. Thus, \(N = \# S^{n} = B^{n}\). In this case, the base is \(B = \# S\) the number of symbols available, an identity which does not apply to other forms of notation for symbolisation than positional notation whereby the indication of place value of a symbol by its position in a permutation allows the recycling of the characters \(^{S}X\) from one place to another in the \(i^{\text{th}}\) word or number \(^{N}X_{i}\).

Extension to Permutations
A permutation is a sequence, not necessarily linearly arranged visually but which may nevertheless be mapped to a linear sequence, in which the position of a symbol or character in the sequence affects the value of the appellational number or word produced by the permutational sequence. Unlike positional notation, the number of symbols available to each position need not be the same, although positional notation is a subcategory of permutational sequences. In the case of general permutations, the number \(N\) of possible numbers standing for words is equal to the product \(\prod_{i}^{n} \text{#}S_i\) of the cardinal numbers of the sets \(S_i\) of symbols available at the places \(i\). The cardinal numbers #\(S_i\) of the sets \(S_i\) of symbols \(^{S}X\) for positions \(i\) may be called subbases \(b_i\) of the number \(N\) of possible word numbers as though it were an overall base. Thus, \[N = \prod_{i}^{n}b_{i}\] However, the total number of symbols available from the sum \(\sum_{i}^{n}b_i\) of all the cardinal numbers \(\# S_i\) provided there are no symbols \(^{S}X_{i}\) common to any of the sets \(S_i\) may be interpreted as a base \(B\) in the sense that it is analogous to the number of symbols available in positional notation and it is less than the number \(N\) of words that may be formed out of it. If there are symbols \(^{S}X_{i}\) common to any of the sets \(S_{i}\), the base \(B\) would be the cardinal number of symbols of the union of the sets \(S_i\) without counting any of the symbols \(^{S}X_{i}\) more than once. I may represent this as $$B = \# \cup S_i$$
Limitations in Zeroless Notational Systems Under Constraints
Consider the Braille system of writing. For each permutation composing a glyph \(^{N}X_i\), there are six positions or places, and in each position there are two possibilities: either a dot or no dot. These are the only two symbols \(^{S}X_i\) and they are common to all positions. Thus, the base of the positions is binary and the number \(N\) of possible glyphs \(^{N}X_i\) from the equation \(N = B^{n}\) given above is \(2^{6} = 8^{2}\).

However, while one of the symbols of the binary base is a dot, the other is no mark. This has the implication that this system does not represent a value of zero in the positions within the glyphs (as opposed to by the glyphs) by any symbol. We may interpret this particular version of a binary symbolic set \(S\) as containing the dot as one element and a null element (normally called the null set) as its second member. That is, \(S = \{ \{\}, \bullet \} \).

The number of useful glyphs in the Braille system of writing under conditions in which a small number of glyphs may appear in a statement is reduced by some of them being equivalent under translational operations or shifting of the glyphs. Hence, the actual number \(N\) of glyphs is less than \(B^{n}\). The amount of information that can be usefully encoded under such a constraint is less than would be expected from the number of elementary symbols and the number of positions within the glyph.

To quantify the efficacy of a base \(B\) and a system of formation of words from its symbols \(^{S}X\) in representing information \(^{N}X\), I introduce a test which I call potency, \(P\). The rationale of the potency \(P\) test or quantification is to find the exponent equivalent to \( \log_{B}{N}\) to which the base \(B\) must be raised to give the number \(N\) of different word numbers \(^{N}X_{i}\). This exponent represents the number of positions or number of times symbols ought to appear in the words \(^{N}X_{i}\). If this hypothetical number of symbols per word \(^{N}X_{i}\) is less than the number \(n\) of symbols \(^{S}X_i\) that actually occur in making up the words \(^{N}X\), then their ratio or fraction \(P\) is taken: \[P = \frac{\log_{B}{N}}{n}\] For example, if there are only 44 = 38⁏ Braille glyphs practical under the translational constraint, then the potency of the system is reduced to \( \frac{\log_{2}{44}}{6} \approx 0.91 \approx\) 0⁏TE.
(To be continued …)

Improving the Potency
A shape or pattern for the arrangement of six positions other than the rectangular one of Braille can produce a better potency. A trigonal arrangement of the same number six of positions can produce fifty or four dozen plus two glyphs that remain distinguishable under translational shifts, while the direction of writing stays the same. The potency of the trigonal arrangement is calculated to be: \[P = \frac{\log_{2}{50}}{6} \approx 0.94 \approx\] 0⁏E3. Hence, under the translational constraint, the potency of the trigonal arrangement is better than the rectangular quadrilateral arrangement even though they use the same base and number of positions.

Caution in Interpretation of the Potency
In the preceding example, the potency test was used for comparing two different arrangements that had the same base and the same number of positions. The potency test can also be used for comparing efficacies with different bases. A couple of remarks about caution in interpretation of the potency score are warranted:

1. The potency score only provides a quantification of how well a system of symbolisation approaches a perfectly exponential system. This means that all perfectly exponential systems will have the same maximum score of unity regardless of how big the base is. To provide a more realistic quantification of how well a system represents information, the potency score ought to be combined with another score that penalises bases by their size. One such adjustment could involve multiplying the potency by a scaled or normalised reciprocal of the radix economy of the base such that the optimal base by size has a score of unity. That is, some power of the potency score could be multiplied by some power of \[\frac{e\ln{B}}{B}\] as an approximation.
   
2. The base of a system of symbolisation is only hypothetically the cardinal number of different symbols in the union of the inventory sets in order to query the maximum number of word or number names that could result from such a number of symbols if they were used in a perfectly exponential way as a comparison for the purpose of devising the potency score.
   

Alternating Permutations
While deliberation in the design of the two-dimensional pictorial arrangement of the positions in a glyph \(^{N}X_{i}\) is one way of overcoming translational indistinguishability among the glyphs, another way for reducing the risk or probability of equivalence under translational shifts that can be applied to one-dimensional arrangements is to have different sets \(S_{i}\) of symbols alternate from one position to another.

Consider permutations of two positions in which one of the positions can be filled by an element or symbol \(^{S_{1}}X_{i}\) from one set \(S_{1}\) while the other position can be filled by an element \(^{S_{2}}X_i\) from the other set \(S_2\). Further suppose that there are no symbols or elements \(^{S}X_i\) common to both sets \(S_{1}\) and \(S_{2}\), as necessary for this strategy of tackling the constraint. Letting the hypothetical base \(B = \#S_{1} + \#S_2\) be the sum of the cardinal numbers of the two sets, the potency may be calculated as: \[P = \frac{\log_{(\#S_{1}+\#S_{2})}{(\#S_{1}*\#S_{2}})}{2}\] Computing these potencies for particular different permutational systems in which the geometric mean \(\sqrt{\#S_{1}*\#S_{2}}\) or square root of the product of the cardinal numbers \(\#S_{1}\) and \(\#S_{2}\) of the two sets \(S_{1}\) and \(S_{2}\) is the same number, or in which the number \(N\) of possible glyphs or words \(^{N}X\) is the same, the potency is found to decrease as the difference \(\#S_{1}-\#S_{2}\) between the cardinal numbers of the two sets increases, as expected because of the increasing deviation from a perfectly exponential scheme. However, this trend does not mean that systems in which the number \(\#S\) of available symbols for both positions is the same will always be more potent than systems in which the cardinal numbers \(\#S_{i}\) available are different. For example, a system in which the number \(N=\#{}^{N}X\) of possible appellations \(^{N}X_{i}\) is apparently a power of base six where \(\#S_{1}=18\) = 16⁏ and \(\#S_{2}=12\) = 10⁏ has a higher computed potency score of \(P \approx\)0.7902 \(\approx\) 0⁏959 than that of \(P \approx 0.7819 \approx\) 0⁏947 for a system with a dozenal power of words where \(\#S\) of both sets is twelve. This is because that senary scheme uses a higher subbase than one in the dozenal scheme, and this has not been penalised in the potency score.
(To be continued …)

References/See also:

• https://en.wikipedia.org/wiki/Radix_economy
  
• https://dozenal.forumotion.com/t52-base-optimality
  

Application of Potency
An alternating permutation of different sets \(S_{i}\) of symbols to form names \(^{N}X_{i}\) may be necessary where elements \(^{S}X_{i}\) within the sets are too similar to each other to be used and distinguished beside each other. For example, Let \(\#S_{1} = H\) be a variable representing the number of horizontal lines in a glyph \(^{N}X_i\). Because of reasons such as translational overlap, the need to subitise the lines, and the need for spacing between groups, it would be better if different groups of lines of the same orientation are not used as different symbols \(^{S}X_i\) within the same glyph \(^{N}X_i\). That is, there should be only one number of horizontal lines in one glyph. The orientation of the lines is analogous to a position in a permutation, in the sense that the angle of the orientation becomes a position along an arc. Likewise, symbols of another orientation may be combined with those of the first orientation but not with each other. Let \(\#S_{2}=V\) be the number of vertical lines composing a glyph word \(^{N}X_{i}\).

To Determine the Most Potent Bases for Alternating Positions
For up to a given number \(V\) of vertical lines that may be selected as elements from one set, we may want to know up to how many \(H\) horizontal lines should be available in order for the potency of the system of symbolisation to be maximised. Equating the partial derivatives \[\frac{\partial P}{\partial \#S_{i}}\] to zero, the relationship \[\left( \frac{1}{H}+\frac{1}{V}\right)(H+V)^{(V/H)}=1\] is derived. For up to a given whole positive natural number \(V\) of vertical lines, the number \(H\) of horizontal lines available for selection can be found such that the formula above is as close as possible to unity. A closer approach to unity indicates a better potency. Thus it is found that for up to seven vertical lines, there should be up to twice as many horizontal lines available. When the number of vertical lines available for permutation is increased beyond seven, the ratio of horizontal lines available to vertical lines available decreases to less than twice. When the number of available vertical lines is decreased to less than seven, the ratio of horizontal to vertical lines available increases to more than duple. There is a minimum number of horizontal lines of about nine when the number of vertical lines is about three. When the number of vertical lines available is two, the number of horizontal lines available in order to give the maximum potency under these constraints is about ten. Thus, if a minimal total number \(B\) of symbols is desired, then their number should not be less than twelve. In these conditions, twelve appears to be the first whole number \(B\) of symbols for which there is a regional optimum of potency while one of the variables is being held constant.

For a given number \(V\) of vertical lines, the number \(H\) of horizontal lines that should be chosen to maximise the potency is approximated by the following hyperbolic equation: \[H=\frac{4}{3}V+\frac{9}{2}+\frac{4/3}{V - 3/2}\] in which the last term on the right hand side may be omitted to give a linear model when the number \(V\) of vertical lines is greater than three. I also produced a model involving a logarithm as follows: \[H=\frac{137}{24}\left(\frac{V+1/2}{\ln\frac{V+1/2}{4/3}}-2 \right)\]
Extension to Three Positions
Consider permutations in which there are three positions to be filled with elements from more than one set where there are no elements common to the sets. We do this because there must be at least two sets with no elements in common in order for the symbolisation not to be trivially exponential. Consequently, the sets alternate with the positions such that one position is filled by selecting at any particular time one symbol from up to \(V\) symbols of one set \(S_{2}\), while the other two positions are to be filled by selecting symbols from up to a number \(H\) extracted from other sets \(S_{1}\) and \(S_{3}\). The set \(S_{3}\) can be a subset of \(S_{1}\) while maintaining the alternation, provided the position for set \(S_{2}\) is between those of the other two sets in an open permutation. For maximisation of the potency, it turns out that we need only consider when the sets \(S_{1}\) and \(S_{3}\) are equivalent and hence have the same number \(H\) of elements. Now, the total number of symbols is the sum of numbers \(H\) and \(V\) of the two different sets without counting \(H\) twice. For such an arrangement, the potency becomes \[P=\frac{\log_{(H+V)}{(VH^{2})}}{3}\] In this case, to maximise the potency for a given number \(H\), the number \(V\) should be more than \(H\) for \(H=3\), about the same as \(H\) for \(H\) = four or five, and less than \(H\) for \(H\geq 6\). Among other results, I found that when \(H=8\), \(V\) is best as six; when \(H=\) twelve, nine is best for \(V\), but the next best is eight and nearly as good; for \(H=\) the square of four or twice eight, \(V\) is best at eleven or twelve; when \(H\) is twice nine, \(V\) should be a dozen plus one or a dozen with about the same potency; and when \(H\) is two dozen, \(V\) should be the square of four or twice eight. Of these, the one having \(H\) as twelve and \(V\) as nine produces a number \(N\) of names that is the fourth power of six. Thus, with a favorable potency it is possible to encode four digits of numbers written in base six by permuting twelve symbols of one kind in two positions with nine of another kind between them.

Extension to Variable Numbers of Positions
For systems of symbolisation not to be exponential, consider the filling of two positions with symbols from different sets that do not overlap in elements to be mandatory, while filling of other positions is optional or discretionary. Let the mandatory sets have cardinal numbers \(H\) and \(V\) of elements, while the discretionary sets have \(D_{i}\) cardinal numbers of elements. In glyphs, the \(D_{i}\) may represent numbers of diagonal or oblique lines, for example. The discretionary sets contain the null element or null set which when selected shows no symbol and the position is not counted in the length of the resulting label \(^{N}X_{i}\). I also contrive that the set of \(D_{2}\) is a subset of another of the sets. The potency becomes \[P=\frac{(D_{1}+1)(D_{2}+1)(D_{3}+1)\ln{\left[HV(D_{1}+1)(D_{2}+1)(D_{3}+1)\right]}}{(2+3(D_{1}+D_{2}+D_{3})+4(D_{1}D_{2}+D_{1}D_{3}+D_{2}D_{3})+5D_{1}D_{2}D_{3})\ln({H+V+D_{1}+D_{3})}}\] With this formula, I investigated values for \(H, V, D_{i}\) that produced numbers \(N\) of labels \(^{N}X\) that were powers of simple bases like binary, senary, or dozenal, as well as the potency scores associated with them. It is possible, for example, to construct a system of symbolism in which the number \(N\) of label words or numbers \(^{N}X\) is the fourth power of twelve and the total number of symbols forming them is as many as or fewer than the number of letters in the alphabet. The system can be applied to description of words where H represents true consonants, V represents vowels such that there is only one vowel per word label, and the other sets represent phonemes of intermediate sonority.

Argument for Alternation
While the formulation of potency was intended to be general to any variety of the natures of characters forming the sets available to the positions, specific examples can be used to illustrate arguments for the benefit of alternating sets from which characters are selected from one position to another. For example, consider the particular application where set H contains characters differing by their numbers of horizontal lines and set V contains characters differing from each other by their numbers of vertical lines. It would not be practical to distinguish different characters from the set V of vertical lines placed next to each other, for example III and III placed together as IIIIII, by the compounding of the two reasons of firstly the size of the spaces between them being potentially confused with the size of the spaces within them and secondly the decreasing ease of subitisation, leading to the interpretation ambiguously as a permutation of the characters II and IIII or I and IIIII instead of III and III. Likewise, but to a lesser extent, it would be impractical to distinguish characters of horizontal lines next to each other, in this case because the lines could be perceived as running into each other because of the tendency of perceiving patterns by filling in gaps with continuations of the patterns. For example the digraphic ≡≡ has a slight tendency to be perceived as monographic ≡. On the other hand, the characters in the digraphic word symbol ≡= would be more distinguishable from each other.

Another example would be that monophthong vowels in digraphs would tend to be indistinct without a hiatus between them, and likewise some sets of consonants, especially those of similar place of articulation, would be less easy to keep separated in pronunciation and acoustically without vowels between them. The natures of H and V are not limited to these examples for the sake of argument. For example, H and V could represent sets of numbers of different coloured lights or different types of flowers in a bouquet.

Allowed Polygraphic Forms
To reduce translational error in the interpretation of strings of characters, they may be permuted in confined ways by a chosen rule.

1. Where each position is filled by a monographic character selected without restriction from one set, the resulting permutation is perfectly exponential positional notation. This is prone to translational error whereby numbers that are far apart in size look similar, as seen in scientific notation of numbers of which only the exponent of the base differs.
   
   
2. If the rule is that word symbols must be digraphs of characters from set H followed by characters from set V, then strings of digraphs would have an alternating pattern such as HVHVHV, to be interpreted with grouping of the digraphs as (HV)(HV)(HV). A translational shift of an odd number of positions would lead to a nonsensical interpretation such as H(VH)(VH)V.
   
   
3. If the chosen rule is that the word symbols should all be trigraphic, then the possible pattern for each word symbol ought to be HVH, because adjacent word symbols of the form VHV would be less distinct at the VV boundaries than word symbols of the HVH form would be at their HH boundaries, in the cases of horizontal and vertical lines or consonants and vowels. A translational error in the chains of HVH word symbols would lead to an impermissible permutation, such as VHH or HHV.
   
   
4. The chosen rule for the arrangement of the sets H and V in the positions for polygraphic word symbols of more than three characters cannot avoid adjacency of characters from the same set without splitting the word symbol into smaller polygraphs. For example, a tetragraph as HVHV would in reality be a pair of digraphs as (HV)(HV). The tetragraphic arrangements HVHH and HHVH would lead to characters of the set H being enclosed by other characters of the same set on both sides in chains of the tetragraphs.
   
   
5. Pentagraphs such as HVHHV could be allowed if they are decomposed in interpretation as pairs of polygraphs HVH and HV. The sixth power of twelve for the number of possible word names of the resulting two syllables can be obtained with relatively high potency in a number of ways by this arrangement, for example if the cardinal number of the set H is three dozen and the cardinal number of the set V is eight, or by the cube of three for two positions and the square of four for three positions. ②⓪⁏ squared times the squares of ①⑥⁏ and four is another possibility, though these values can be quite large for sets of vowels or consonants. It is also possible to have the sixth power of eight for the number of possible word symbols or names where the cardinal number of the set H is four squared or twice eight and the cardinal number for the set V is eight, with a relatively good potency.
   
   
6. Hexagraphs as HVHHVH would be the same as pairs of trigraphs HVH.
   

Polygraphic Word Symbols Of Fixed Length
Digraphs
Where characters from two mandatory sets alternating as H and V and no discretionary sets D form the word symbols, the potency score indicated that increasing the numbers of characters available or cardinal numbers of the sets H and V tended to increase the potency, as did reducing the difference between the two cardinal numbers of the sets H and V, but for a fixed value of the number of available characters in one set there was an optimum number for available characters from the other set resulting in alternating numbers of available characters in the resulting digraphic word symbols. If the number of possible word symbols formed is to be a power of a numerical base, then the product of multiplication of the cardinal numbers of sets H and V must equal the chosen power of the numerical base. Whereas it is easy to allow numbers up to the second power of the numerical base twelve to be represented by digraphic word symbols of twelve characters available from each of the sets H and V, as opposed to letting the cardinal numbers of the sets H and V be rather alternating as say four squared for one and nine for the other, it is less practical for numbers up to the third power of the numerical base twelve to be represented by digraphs, because the cardinal numbers of available characters required for the two sets would be impractically large such that they would become less easy to distinguish. While the third power of twelve could be obtained as the product of multiplication of two factors in the following ways:

• four dozen times three dozen, or ④⓪⁏ times ③⓪⁏
  
• ④⑥⁏ times ②⑧⁏
  
• six dozen times two dozen, or ⑥⓪⁏ times ②⓪⁏
  
• eight dozen or ⑧⓪⁏ times ①⑥⁏
  

all of these numbers are too large for example to be numbers of available vowels that are readily distinguishable without causing the identities of their adjacent consonants to merge with each other. Hence, in order to achieve the third power of twelve, a more elaborate arrangement than the digraphic would be required.

For this reason, the discretionary sets among two mandatory sets were considered. These allowed twelve cubed word symbols, but decreased the potency. A higher power of twelve can be achieved by the addition of a terminating mandatory character of the set H type.

Trigraphs
A terminal character of set H can be added to a digraph of form HV to produce a trigraph HVH while maintaining an identical potency because the total number of available symbols is the same and
\[ \frac{\ln{B^{3}}}{3\ln{2B}} = \frac{\ln{B^{2}}}{2\ln{2B}}\]
This enables the third power of the numerical base twelve to be reached with twelve characters available to each position with a relatively good potency (of which I was aware before the opening post in this topic last year). However, a better potency is obtained by reducing the number of available characters from set V of the medial position from twelve to nine, leading to a number of word symbols or names that is the fourth power of six, as mentioned previously above, rather than a power of twelve. Thus, according to the potency score, it is possible to increase the potency by decreasing the number of different characters or numerals and having alternating numbers of characters available in the positions rather than maintaining a single numerical base of division consistently at each position.

The fourth power of twelve can be reached with a relatively high potency by trigraphic word symbols of the form HVH where the cardinal number of the set H is four dozen and the cardinal number of the set V is nine, or by three dozen for H and four squared for V. In the case of consonants and vowels, these cardinal numbers are at the high end of what is convenient for distinguishability, but not impossible.

To find where maximum potencies for the trigraphs of form HVH are, the equation to zero of the partial derivative
\[\frac{\partial P}{\partial V}\]
of the potency with respect to V keeping H constant was found to produce the relationship
\[\left(\frac{1}{VH}+\frac{1}{H^{2}}\right)(V+H)^{(H/V)}=1\]
For a fixed value of H no less than four, the optimum value for V is modelled by this linear equation:
\[ V \approx \frac{11}{18} H + \frac{14}{9} \]
The following hyperbolic model was produced which can be applied also for H less than four:
\[ V \approx \frac{5}{8} H + \frac{9}{8} + \frac{21/16}{H - 1} \]
I also made a model containing a logarithm as follows:
\[ V \approx \frac{8H}{3\ln{H}}-\frac{61}{16} \]

A compromise between the optimum cardinal numbers of H and V for the digraphic and trigraphic arrangements is modelled by this linear equation:
\[ H \approx \frac{41}{27}V + 1 \]
For high values of V and H, a directly proportional ratio of H to V of three to two is a suitable rough compromise to aim for.

Discretionary Characters
Where there is only one mandatory set, say H, and one discretionary set, say D, for the formation of name symbols, the potency is given by the formula:
\[P=\frac{(D+1)\ln{\left[H(D+1)\right]}}{(1+2D)\ln{(H+D)}}\]
Where there are three mandatory positions and up to three discretionary positions, and where the characters of one of the discretionary sets are the same as those of another discretionary set, the following formula applies and was used to investigate potencies:
\[P=\frac{(D_{1}+1)(D_{2}+1)(D_{3}+1)\ln{\left[H^{2} V(D_{1}+1)(D_{2}+1)(D_{3}+1)\right]}}{(3+4(D_{1}+D_{2}+D_{3})+5(D_{1}D_{2}+D_{1}D_{3}+D_{2}D_{3})+6D_{1}D_{2}D_{3})\ln({H+V+D_{1}+D_{3})}}\]
Several possibilities for reaching the fourth power of twelve without such large cardinal numbers for the sets were found, though the potency was reduced by the discretionary sets. One way was to add two discretionary sets to the trigraphic arrangement of twelve for H and nine for V.

Pentagraphs
If a terminal position is appended to a digraph of form HV by allowing a character there to be chosen from the same set H as starts the digraph to form a trigraph, then it could not be possible to determine the boundary between a string of such trigraphs and digraphs, as HVHHV could be parsed as either a trigraph followed by a digraph as (HVH)(HV) or as a pair of digraphs (HV)(HV) if the same character from set H happens to appear at the junction between the word symbols, because a doubled identical character of form HH could be misinterpreted as a single instance of that character of set H. This has the implication that a terminal character of a trigraph cannot be discretionary if it is of the same set as the initial character of the word symbol, but must be mandatory if trigraphs are allowed and designed such as to maintain the potency by repetition of characters of the same set H in different positions. If it is desired that both digraphic and trigraphic word symbols are allowed to appear freely in any order, then the terminal set of characters must be different from the other two sets, but this would decrease the potency.

Therefore, if word syllables of both digraphic and trigraphic forms are to appear by selecting characters from just two sets H and V, then they must appear in a prescribed order, such as in the pentagraphic form HVHHV or one of its cyclic permutations, and all word symbols must have this predictable form and constant length so as to allow recognition of the characters separately in each position. Even if two identical characters of the same set H appear beside each other, they would have to be interpreted as a double instance of the character rather than one instance of the character in order for the word symbol to conform to the prescribed length.

The potency P of the pentagraphic word symbol of form HVHHV is
\[ P = \frac{\ln{(H^{3}V^{2})}}{5 \ln{(H+V)}} \]
Equating to zero the partial derivatives
\[\frac{\partial P}{\partial H}\]
and
\[\frac{\partial P}{\partial V}\]
of the potency P with respect to the numbers of characters in each set separately while holding the number of characters in the other set constant produces the following identities:
\[ \frac{(H+V)^{3(H+V)/H}}{H^{3}V^{2}}=1 \]
and
\[ \frac{(H+V)^{2(H+V)/V}}{H^{3}V^{2}}=1 \]
For the first of these, the number of characters H in terms of V can be approximated by the following hyperbolic function:
\[ H \approx \frac{20}{9} V + \frac{173}{27} + \frac{11/2}{V - 1} \]
while from the second relationship, the number of characters V in terms of the number of characters H can be approximated by the hyperbolic function:
\[ V \approx \frac{8}{9} H + \frac{5}{3} + \frac{2}{H - 1} \]
These relationships were used to find cardinal numbers of the sets H and V that produced potencies larger than potencies from numbers of similar size for the pentagraphic form HVHHV. Notable from the first relationship is the system with nine for V and three cubed for H, producing a ternary power with relatively high potency. Also, a decimal power is produced from eight for V and the square of five for H with relatively high potency. From the second relationship, notable is the system where the numbers of available characters for both H and V are the square of four, producing a binary power with a relatively high potency and the same potency as a digraph with that number of characters available in each position. The pentagraphic form is another way of increasing the number of possible word symbols without decreasing the potency.

A comparison of systems of different forms of fixed length ranked by potency can now be presented in the following table:

Form and Position Cardinal Numbers					Potency	N as power of a numerical base
H1	V1	H2	H3	V2
54	36	54	-	-	0.85644	18^4
36	16	36	-	-	0.83852	12^4
24	16	24	-	-	0.82489	96^2
48	9	48	-	-	0.81948	12^4
16	16	16	16	16	0.8	16^5
24	18	18	24	16	0.79779	12^6
27	9	27	27	9	0.797091	3^13
27	16	16	27	16	0.79280	12^6
25	8	25	25	8	0.79025	10^6
18	12	-	-	-	0.79020	6^3
16	9	16	16	9	0.78985	24^4
36	8	36	36	8	0.78799	12^6
16	8	16	16	8	0.78518	8^6 = 64^3
12	9	12	-	-	0.78469	6^4 = 36^2
12	12	-	-	-	0.78190	12^2
16	9	-	-	-	0.77198	12^2
24	9	-	-	-	0.76866	6^3
36	6	36	36	6	0.76701	6^8 = 36^4
16	8	-	-	-	0.76337	2^7
18	8	-	-	-	0.76269	12^2
9	9	-	-	-	0.76019	3^4 = 9^2
9	8	9	9	8	0.75889	6^6 = 36^3
8	8	-	-	-	0.75	8^2 = 64
24	6	24	24	3	0.73060	12^5
24	6	-	-	-	0.73060	12^2
6	8	6	6	8	0.72254	24^3
6	6	-	-	-	0.72106	6^2
6	4	6	-	-	0.71945	12^2
36	6	-	-	-	0.71907	6^3

(To be continued…)