3. Periodicity and symmetry in linear dashed lines

Prerequirements:

In this post we’ll talk about two fundamental properties of linear dashed lines: periodicity and symmetry.

Periodicity and number of dashes in a period

In Table 4 in the post From prime numbers to dashed lines we saw that, in the representation of a linear dashed line, there is a certain outline that repeats itself infinitely many times. This is expressed mathematically saying that a linear dashed line is periodical, in the sense of the following definition:

Periodical dashed line

Let T be a dashed line. If there exists a positive integer l such that, for all i, the i-th and the i+l-th dash of T:

  • have values that differ by a fixed amount M, that depends only on T
  • are in the same row

Then

  • T is said periodical
  • M is said length of the period
  • Any set of M consecutive natural numbers is said period
  • l is said number of dashes of the period

With reference to the graphical representation of a dashed line, this definition states that we can find any dash of a periodical dashed line, generally the i-th dash, again in the same row after M columns, where the number M is a fixed quantity, characteristic of the particular dashed line we took. Of course the dash placed M columns after the i-th one, in the same row, will be the (i+l)-th, for some positive integer l. According to the definition, this l is constant too and typical of the particular dashed line.

All the dashed lines seen in the post From prime numbers to dashed lines, apart from those of Tables 1, 10 e 11, are periodical. It’s not by chance: these dashed lines are in fact all linear, and it can be proved that a linear dashed line is periodical. We already mentioned this property concerning the dashed line (2,3,4,5) (Table 4), but now we’ll see in detail the underlying mechanism, in the case of the dashed line (2,3). Let’s write in the graphical representation, in place of the dashes, as in Table 13, their ordinal numbers:

Table 14: graphical representation of the linear dashed line (2, 3), where the ordinal numbers of the dashes are written and three periods are highlighted
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 3 4 6 8 9 11 13 14 16
2 5 7 10 12 15

In this table we can see that the same arrangement of dashes repeats itself every 6 columns, 6 because it’s the least common multiple of the dashed line components, that are 2 and 3. In particular, taking columns 6 by 6 as highlighted in Table 14, the arrangement that repeats itself is the following:

Table 15: Arrangement of dashes that repeats itself in Table 14, taking columns 6 by 6 starting from column 0

Also starting from another column, and again taking columns by groups of 6 consecutive ones, we have a repeating outline. For example, if we disregard column 0 and consider the column groups 1-6, 7-12, 13-18, etc., the repeating outline is the following:

Table 16: Arrangement of dashes that repeats itself in Table 14, taking columns 6 by 6 starting from column 1

We would have a similar situation starting with any other column, still taking columns by groups of 6 consecutive ones: in any case there will be a repeating arrangement of dashes. The so obtained arrangements are different from each other (as you can see by comparing Tables 15 and 16) but we can note that they contain the same number of dashes, 5 in the example we are considering. The reason for this is that, however we take 6 consecutive integers, three of them are divisible by 2 and two of them are divisible by 3: for example in the set \{1,2,3,4,5,6\} the numbers divisible by 2 are 2, 4 and 6, and the ones divisible by 3 are 3 and 6: this exactly coincides with what’s shown in Table 16. Summarizing, we have that:

  • The dashed line (2,3) has an outline repeating every 6 columns, no matter what the starting column is
  • Every so obtained outline contains 5 dashes, no matter what the starting column is

Going further, we can observe what follows. Having divided the dashed line into groups of 6 consecutive columns, as in Table 14, we can start from any dash and consider the fifth next dash: this way we get a dash placed in the next group of 6 columns, but in the same position with respect to the repeating arrangement.
For example, with reference to Table 14, starting from the first dash, the dash (1,1) with value 2 placed in the red part, and going ahead by 5 dashes, we’ll get to the dash (1,4) of value 8 in the green part. It’s in the first row too and, with respect to the outline shown in Table 15, it corresponds to the third column too:

Table 17: Repeating outline in Table 14, taking columns 6 by 6 starting from column 0, where the column corresponding to columns 2, 8, etc. of Table 14 is highlighted.

But two dashes placed in two consecutive outline repetitions (like the red one and the green one in Table 14) and that, with respect to it, are placed in the same column (Table 17), certainly have values that differ by the length of the outline itself, 6 in this case. In fact the dash (1,4) has value 8 and the dash (1,1) has value 2 = 8 – 6.
We could also repeat the same argument in the opposite verse: if we start from a dash and move 6 columns (the length of the outline) to the right, we’ll get to a dash placed in the same position with respect to the outline, and 5 dashes after the starting dash, where 5 is the number of dashes contained in the outline.

Now we can understand in what sense the dashed line (2,3) is periodical according to Definition T.5:

  • Every group of 6 consecutive columns, starting from any column, is made up of the same outline (arrangement of dashes)
  • Starting from any dash t, the fifth next dash u is placed in the same position as the starting one, with respect to the outline
  • The values of t and u differ by the length of the outline

So, referring to the definition, the outline is a period, M = 6 is the length of the period, and l = 5 is the number of dashed of the period.

Examining in depth what exposed above and generalizing, the following properties can be proved:

Representation of a periodical dashed line

In the representation of a periodical dashed line having period length M, for all integers i \geq 0, column i is identical to column i + M.

Number of spaces in a period

In the representation of a periodical dashed line, any period has the same number of spaces.

Linear dashed lines are periodical

A linear dashed line (n_1, n_2, \dots, n_k) is periodical, with period length M = \textrm{MCM}(n_1, n_2, \dots, n_k) and number of period dashes l = \frac{M}{n_1} + \frac{M}{n_2} + \dots + \frac{M}{n_k}.

The proofs will be added in the future.

The formula for computing the number of dashes in a period is of particular importance, because it ofter will recur in the formulas of dashed line theory, that we’ll see in the next posts. We are particularly interested in a specific case of this formula, when dashed line components are two by two coprime. In this case their least common multiple coincides with their product: M = n_1 \cdot \ldots \cdot n_k. So the following Corollary of Property T.4 is obtained:

Number of dashes in a period of a linear dashed line with two by two coprime components

Let T = (n_1, \ldots, n_k) be a linear dashed line the components of which are two by two coprime. The number of dashes in any period of T is given by the formula

\frac{n_1 \cdot \ldots \cdot n_k}{n_1} + \ldots + \frac{n_1 \cdot \ldots \cdot n_k}{n_k} = \sum_{i=1}^k \frac{n_1 \cdot \ldots \cdot n_k}{n_i} \tag{1}

In particular, for the orders k \leq 3, inverting the order of summation terms, the following formulas are obtained:

  • If k = 1: n_1
  • If k = 2: n_1 + n_2
  • If k = 3: n_1 n_2 + n_1 n_3 + n_2 n_3

We can note that the polynomials generated by (1) are symmetric: in each of them we can exchange in any way two or more variables, always obtaining the same polynomial. For example, in the polynomial n_1 n_2 + n_1 n_3 + n_2 n_3, putting n_1 in place of n_2​, n_2​ in place of n_3​ and n_3​ in place of n_1​ we obtain the polynomial n_3 n_1 + n_3 n_2 + n_1 n_3 that, taking into account the commutativity of product and sum, is equivalent to the initial polynomial.
But the polynomials generated by (1) are not ordinary ones: they are called elementary symmetric polynomials (v. Definitions and symbols) and they have great importance in algebra. The polynomial obtained above for a certain value of k is denoted with the notation \sigma_{k-1}(n_1, \ldots, n_k).

Symmetry

Not only linear dashed lines are periodical, but they are also symmetrical, in the sense that, if M is the period length, taken the set of the M+1 consecutive columns \{0,1,\ldots,M-1,M\} (that is the period starting with column 0, plus another column), the last column of this set coincides with the first one; the second to last coincides with the second one, and so on.
We can easily view this property by resuming the dashed line (2,3) of Table 14. In this case the period length is \textrm{LCM}(2, 3) = 2 \cdot 3 = 6, and we can observe the symmetry in the set of columns \{0,1,2,3,4,5,6\}. In fact, column 0 coincides with column 6, column 1 coincides with column 5, column 2 coincides with 4; finally column 3, that obviously coincides with itself, works as center of symmetry of the considered dashed line portion (this center of symmetry exists because the number of considered columns, 7, is odd; otherwise there would still be a symmetry, but without a central column):

Table 18: graphical representation of the dashed line (2, 3), where symmetrical columns of the set \{0,\ldots,6\} are highlighted with the same colour
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

A symmetry of this kind cannot be observed in any set of consecutive columns, for example there is not symmetry in columns \{1,2,3,4,5,6,7\} or \{0,2,3,4,5,6,7,8\}. However there is symmetry in other sets of columns, by virtue of periodicity. For example, again with reference to the dashed line (2,3), we know that columns from 0 to 6 coincide, in an orderly fashion, with those ones from 6 to 12, so this portion of the dashed line has the same symmetry:

Tabella 19: graphical representation of the dashed line (2, 3), where symmetrical columns of the set \{6,\ldots,12\} are highlighted with the same colour
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Likewise, we can observe symmetry in any set of columns of the kind \{k \cdot M, \ldots, (k + 1) \cdot M, for all k \geq 0. But we can do a step further, applying periodicity for extending the portion of considered symmetrical columns. In fact:

  • Column 0 is symmetrical to column 6 with respect to the range of columns from 0 to 6, but column 6, by periodicity, coincides with column 12
  • Column 1 is symmetrical to column 5 with respect to the range of columns from 0 to 6, but column 5, by periodicity, coincides with column 11
  • Column 2 is symmetrical to column 4 with respect to the range of columns from 0 to 6, but column 4, by periodicity, coincides with column 10
  • Column 3, by periodicity, coincides with column 9
  • Column 8 is symmetrical to column 10 with respect to the range of columns from 6 to 12, but column 10, by periodicity, coincides with column 4
  • Column 7 is symmetrical to column 11 with respect to the range of columns from 6 to 12, but column 11, by periodicity, coincides with column 5
  • Column 6, by periodicity, coincides with column 0

Taking all these observations as a whole, we can obtain the symmetry of a wider range of columns, from 0 to 12:

Table 20: graphical representation of the dashed line (2, 3), where symmetrical columns of the set \{0,\ldots,12\} are highlighted with the same colour
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

In the same way we can see that any set of columns between two multiples of the period length M, is symmetrical. We state the property that linear dashed lines are symmetrical, in this more general form:

Linear dashed lines are symmetrical

Given a linear dashed line having period length M, taken any range of consecutive columns between two multiples of M included, they are symmetrical, in the sense that the first of the considered colums coincides to the last one, the second coincides with the last second one, and so on. In other terms, for all h \geq 0 and for all k \gt h, column h \cdot M + i coincides with column k \cdot M - i, for all 0 \leq i \leq (k - h) \cdot M.

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