CC BY
11
Computer tools in education, 2017 № 4:14-28
http://ipo.spb.ru/journal
ON COMPUTER MODELING OF FINITE-GENERATED FREE PROJECTIVE PLANES
Gogin N. D., Myllari A. A.1
1St. George's University, Grenada, West Indies
Abstract
This paper treats computer modeling of the process of constructing free projective planes — more precisely, to algorithmically finding their successive incidence matrices; and also to considering some numerical characteristics of these matrices. Matrix and bilinear forms approaches are used to study the growth rate of the number of new elements (points, lines) during step-by-step process of constructing projective plane starting with the Hall n4 configuration. It appears that the number of new elements grows asymptotically as a double exponent (linear on log(log) scale.) Rough estimate from above also gives double exponential growth rate.
Keywords: free projective planes, finite geometries, combinatorial design.
Citation: N. D. Gogin & A. A. Myllari, "On Computer Modeling of Finite-generated Free Projective Planes", Computer tools in education, no. 4, pp. 14-28, 2017.
1. INTRODUCTION
W. W. Sawyer in his Prelude to Mathematics [1] writes: "Projective geometry is one of the most beautiful parts of elementary mathematics.
For the professional mathematician it is undoubtedly an essential part of one's education. One does not need to go very far with it; the value of a detailed study of it is doubtful, except for the specialist. But the basic patterns of projective geometry can be traced in many other branches of mathematics; they serve to guide and to unify."
The subject of this paper is free and finite projective planes, part of the vast area of modern combinatorics, called the theory of combinatorial designs. Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry [2]. Applications of combinatorial design theory can be found in many areas including finite geometry (finite affine and projective planes, Mobius or inversive planes, etc.), tournament scheduling, experimental design, lotteries, mathematical biology, algorithm design and analysis, networking, finite groups theory, and cryptography. We address interested readers to the previously cited article in Wikipedia and references therein. Combinatorial designs have a long history: for example, the magic square of order three, the so-called Lo Shu Square, dates at least to 650 BC; the oldest image of this square was found on a tortoiseshell dated 2200 BC
(according to legend the Chinese Emperor Yu observed the magic square
r 4 9 2 '
3 5 7 ,816,
on the back of a divine tortoise [11].) Combinatorial design methods evolved along with the general growth of combinatorics from the 18th century, for example, from the studies of Latin squares and the famous "36 officers problem", which goes back to Leonard Euler (1782) [11]. Today, one can see many people solving Sudoku puzzles — actually, they are solving a classic combinatorial design problem.
Classical subjects of combinatorial design theory include balanced incomplete block designs (BIBDs), symmetric BIBDs, Hadamard matrices and Hadamard designs, difference sets. Other combinatorial designs are related to or have been developed from the study of these fundamental ones.
Let us give for the sake of completeness, the definition of BIBD (balanced incomplete block design), or (b, v, r, k, A)-configuration [11]. Let X be a finite set of v elements. A balanced incomplete block design (or simply block design) is a collection B of b subsets (blocks) of X, such that every block has the same number k of elements, each pair of distinct elements appear together in the same number A of blocks, where k < v - 1, A > 0, and any element of X is contained (replicated) in the same number r of blocks.
It follows immediately from the definition that r(k -1) = A(v - 1) and bk = vr.
A symmetric balanced incomplete block design (SBIBD), (v, k, A)-configuration is a BIBD in which the number of elements equals the number of blocks (v = b). They are the single most important and well studied subclass of BIBDs.
A finite projective plane of order n is SBIBD with parameters v = n2 + n + 1, k = n + 1, A = 1.
The theory of combinatorial designs in general and of finite geometries in particular abounds with a mass of unsolved problems that are difficult to be investigated even with modern methods of combinatorial mathematics. This also applies to the theory of projective planes (see for example the "Prime-power hypothesis for the orders of the finite projective planes" below). In particular, no sufficiently developed general theory of construction and the structure of finite projective planes has been created to date.
In view of this, it seems quite natural that in an effort to create such a theory, mathematicians turned to already known analogous constructions usually called "free objects" of the theory in question. In our case, we are talking about "free projective planes", which being infinite themselves, can shed light on problems associated with finite projective planes.
Of course, the study of free projective planes is also of great interest by itself.
Free projective planes were first introduced by M. Hall in his fundamental paper [3] where he considered their basic properties. Since then, these planes have become the subject of constant interest of mathematicians studying abstract algebraic structures, group theory and their representations, and so on [4, 5, 7? ,8]. There are also good surveys which one can use to get acquainted with the basic concepts and achievements of the modern theory of combinatorial geometries, for example, [6, 10-12]. As a general introduction to the projective geometry, one can use e.g. [13-15].
This paper is devoted to computer modeling of the process of constructing free projective planes — more precisely, to algorithmically finding their successive incidence matrices; and to considering some numerical characteristics of these matrices.
Remarks about notations: If A is a (non-empty) matrix then dim1(A) (resp. dim2(A)) is a number of its rows (resp. columns); [A\i,j means its element at the entry (i, j); Ai (resp. Aj)
means i-th row (resp. j-th column); diag(A) for a square matrix A means column-vector of its diagonal elements; Total[A] is a sum of all elements in A. Moreover, we treat binomial coefficient x and differential operators (derivatives, Laplace operator) as listable functions.
ni,j is a column-vector with "1"-s only in two different positions i and j and all the rest components equal to "0"-s.
As a rule we do not show the matrix format explicitly unless it is not clear from context. E denotes identity matrix; J is a square constant matrix of (only) "1"-s; J * = J - E; {} denotes empty matrix; <, ) means Euclidean scalar product; for a matrix A and real a we define a product a • A as follows:
A o B denotes the element-wise (Hadamard) product of matrices with the identical formats. If A and B are matrices having appropriate formats then A| u B(resp. A u B) denotes a concatenation of A and B from the right (resp. from below) providing A| u {} = A u {} = A.
In this section we mostly follow the terminology and definitions of [6]. Definition 1. A configuration (or a partial plane [1]) is a pair n = (P, L) where P is (nonempty) set of points and L is a family of subsets of P called lines under the condition that the following axiom is valid:
C1: Any two different points are incident with no more than one line. Axiom C1 implies
C2: Any two different lines are incident with no more than one point in common.
As a rule in this paper we shall be interested only in the case of finite sets P. Examples 1.
1. Desargues' configuration directly related to the Desargues' theorem (a classic example of the projective theorem, completely independent of measurement) is well-known (see, e.g. [1], [6]): Mark a point O, draw the three lines OA, OB, OC. Points A, B, and C can be anywhere on these lines. Also choose any three points A', B', C', A! on OA, B' on OB, C' on OC. Join AB and A'B'. These two lines intersect in point F. In the same way, AC and A'C' intersect in point E, BC and B'C' intersect in point D.
Desargues' theorem for the usual real projective plane claims: points D,E, and F lie on a straight line (see Fig. 1).
Desargues' configuration consists of 10 lines, each incident to 3 points, from the other side, there are 10 points, each incident to 3 lines. It has a strong symmetry: any of these 10 points could be marked as O, there always will be a way (actually, several ways) to mark other points so that the statement of the theorem remains true. There are 120 different ways of putting in the letters on the picture without any changes in the printed statement being necessary [1].
2. Another classic example of the projective theorem, completely independent of measurement is Pappus theorem. Pappus configuration we get by taking two lines and choosing three points A, B, C on one line and three points A', Band C' on another line (points should be different from the intersection point of this two lines.) Connect A with B' and Cconnect B with A' and C'; connect C with A' and B'. Let us denote intersection of lines AB' and A'B by D, intersection of lines AC' and CAr by E, intersection of lines CAr and C' A by F.
Pappus' theorem: Points D, E, and F are collinear (see Fig. 2.)
a A, if a ± 0 {}, if a = 0.
2. PRELIMINARIES
Pappus' configuration consists of 9 lines, each incident to 3 points, from the other side, there are 9 points, each incident to 3 lines (cf. [1, 6]).
3. If in Definition 1 L = 0 and |P| = m, m > 0 is an integer, then we have a pure m-points configuration.
4. If L consists of all pairs {a, b}, a, b e P, a + b then n = (P, L) is a full graph on m vertices.
5. Let nm = (P,L), m > 4 be a configuration with |P| = m and only one line A, (i.e. L = {A}) where | A| = m-2. This means that all points besides two of them lie on the (unique) line A. These configurations are called standard [10] or Hall configurations and were first introduced by M. Hall in his fundamental paper [3], p. 237.
Definition 2. Configuration n = (P, L) is called a projective plane, if in axioms C1 and C2 the words "...with no more than one..." are changed by "... exactly one...", i.e. in n = (P,L) the following axioms are valid:
P1: Any two different points are incident to exactly one line;
P2: Any two different lines are incident to exactly one point in common; and in addition the axiom
P3: There exist 4 different points such that no three of them are collinear; in order to exclude some degenerate configurations (cf. [10]).
The following simple statements can be easily proved for a finite projective plane [4]:
A) Every line is incident to exactly n + 1 points;
B) Every point is incident to exactly n + 1 lines;
C) |P| = |L| = N = n2 + n + 1.
The number n is called the order of the finite projective plane. Example 2. Fano plane: for n = 2 we obtain an example of a "smallest" (nondegenerate) projective plane, called the Fano plane. This plane contains 7 = 22 + 2 + 1 points and 7 lines, each line contains 3 = 2 + 1 points and through each point pass 3 lines (Fig. 3).
Figure 3. Fano plane — finite projective plane of order two
"Prime-power hypothesis for the orders of the finite projective planes" claims that always n = pv for some prime p. To date, this hypothesis remains unproved.
Definition 3. If n = (P, L) is a finite configuration with |P | = m and |L| = l, l > 0 then the incident matrix of n is defined as l x m 0-1-matrix A = (a;,j) where
ai, j =
11, if point j is incident with line i I 0, if point j and the line i are not incident
(1)
in some chosen (and fixed) numerations of sets P and L. Example 3.
1. Incident matrix of Desargues' configuration (with proper numbering of points O, A, B, C, A', BCD, E, F) is
0 1 1 1000000^ 1000100100 1000010010 1000001001 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0
010000001
0
0 0 1 0 0 0 0 1 00010001
1
We leave as an exercise for the reader to find corresponding numberings for Fig. 1.
2. Incident matrix of Pappus configuration with ordering points A, B, C, A', BCD,E, F (Fig. 2) is
' 1 1 1 0 0 0 0 0 0 A 0 0 0 1 1 1 0 0 0 100010100 1 0 0 0 0 1 0 0 1 010100100 0 1 0 0 0 1 0 0 1 001100010 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1
3. Incident matrix of the Fano plane (with proper numbering of points A, B, C, D, E, F, G) is cyclic:
' 0 110 10 0 ^ 0011010 0001101 1000110 0 1 0 0 0 1 1 1010001 1101000
Again, we leave as an exercise for the reader to find corresponding numbering for Fig. 3. General properties of the incident matrices are as follows:
a. The i-th row Ai of incident matrix indicates all points incident to the i-th line and
Total[Ai] = Y.ai,j =Z a\j
(2)
j=i j=i = (Ai, Ai) = (number of points on the i-th line)
whereas for i + k the scalar product (A{, Ak} is 0 or 1 according to axiom C2.
b. Dually, the j-th column Aj of incident matrix shows all lines incident to the j-th point
and
Total[Aj] = £ at,j = £ a2 f
(3)
i=1
i=1
= <Aj, Aj ) = (number of lines incedent to the j-th point)
whereas for j + k the scalar product <Aj, Aj} is 0 or 1 according to the axiom C1.
c. So, the i-th diagonal element of the product AAT equals (number of points on the i-th line), whereas the elements outside the diagonal are 0 or 1. Of course, mutatis mutandis this is valid also for AT A. Obviously
Tr ( AAT ) = Tr ( ATA) = Total ( A)
(4)
d. If all the outside-diagonal elements in AAT (resp., AT A) are equal to 1, we say that configuration is line-wise ample (resp. point-wise ample).
Clearly, if n = (P, L) is a projective plane of order n then it is both point-wise ample and line-wise ample and its incident matrix is a square N x N 0-1-matrix such that
AAT = AT A = nE + J
(5)
(cf. for example, [4]).
Example 4. These properties can easily be checked with matricies from the Example 3. For example, for the incident matrix of the Fano plane (n = 2) we have
AAT
'311111 3 1111 3 111 13 11 113 1 1113 1111
3
while for the incident matrix of the Pappus configuration we have
AAT =
0 1 1 1 1 1 1
CO 1 1 1 1 1 1
1 3 1 1 0 0 1
1 1 3 0 2 0 1
1 1 0 3 1 1 0
1 0 2 1 3 0 1
1 0 0 1 0 3 1
1 1 1 0 1 1 3
0 1 1 1 1 1 1
3
3. FREE PROJECTIVE PLANE GENERATED BY CO N FIG U RATIO N
Let n0 = (P0, L0) be some (initial) configuration. The free projective plane generated by n0 is defined by the following process:
1. Let ni = (Pi,L1) be a new configuration where L1 = L0 and Pi = P0 u vP0
vP0 = {(a)(b)|a, b e L0, a and b are not incident in n0} (6)
1.e. every pair of non-incident lines defines a new point named (a)(b) which is "intersection" of lines a and b. Evidently ni is line-wise ample.
2. Let n2 = (P2, L2) be a new configuration where P2 = P1 and L2 = L1 u vL1
vL1 = {(a)(b)|a, b e P1, a and b are not incident in n1} (7)
i.e. every pair of non-incident points a and b defines a new line named (a)(b) which "connects" points a and b. Evidently n2 is point-wise ample.
Iterating this construction we get a sequence (finite or infinite) of configurations {n0,n1;n2,n3,n4,n5,...,nr,...} in which for r even we add new points to nr, as in item 1 and for r odd we add new lines to nr as in item 2 and get next configuration nr+1, r > 0. Proposition 1 (see [6]). If n0 contains 4 different points no three of which are collinear then n = fr (n0) = U^0 n^ is a projective plane.
This plane is said to be the free projective plane generated by n0. Remarks:
1. If an initial configuration n0 is finite and has isolated ("empty") point(s) (resp. "empty lines") then after the first (resp. "second") step of the above algorithm such point(s) (resp. "lines") will vanish, so in order for the computer realization of the algorithm to be implemented correctly, we must always require that the initial incident matrix (and hence all the next) does not contain zero-columns (resp. "zero-rows").
2. The construction of "names" for new points/lines in the above definition gives rise to attempts to consider free projective planes as commutative but not associative universal algebras [? ].
Example 5.
1. If n0 is a projective plane then evidently fr (n0) = n0.
2. If |n0| = 3 and |L0| = 0 then fr (n0) is called a "projective plane of order n = 1" (see Definition 2, p.1) and it is a plane over the field of one element (Fig. 4, left). This plane is referred to as "degenerated" because Axiom P3 evidently is not valid for it. Its incident matrix is cyclic.
The following theorem of M. Hall (see [3]) explains the importance of Hall configuration n4:
Figure 4. Projective plane of order n = 1 (left) and its incident matrix (right).
1) Let n0 is any non-degenerate configuration but not a projective plane. Then fr(n0) contains fr (n4) as a subplane. Moreover, such plane is never desarguesian.
2) A fr (nm), m > 4 contains fr (nm+1).
Everywhere in what follows we deal only with the Hall configuration n4, i.e. fr(n4) = {n4} r=0,1,2,...» that is "free equivalent"(see [3]) to a pure configuration on 4 points, i.e., a full graph with 4 vertices.
4. MATRIX APPROACH
According to what was said at the end of previous section we begin with configuration n0 = n4 (which is zero-step, s = 0, of our algorithm) with incident matrix
A0 =
f 0 0 1 1
0 1 0 1
0 1 1 0
1 1 0 0
1 0 1 0
1 0 0 1
which corresponds to the configuration 4 from Example 1 with m = 4. This configuration (tetrahedron) is shown below on Fig. 5 (left).
Evidently here dim 1(A0) = A0 = 6, dim2(A0) = P0 = 4. Since
' 2 110 11 A
Ao AT =
12 110 1 112 110 0 112 11 10 112 1 110 112
A0TA0 =
CO 1 1 1
1 CO 1 1
1 1 CO 1
1 1 1 CO
this configuration is point-wise ample (any two different points are incident), but is not line-wise ample because exactly 3 pairs of lines, namely 1,4, 2,5 and 3,6, have no points in common.
According to item 1 of the general constructing of fr (n0) at the next step s = 1 we must add to n0 vP0 = 3 new points, namely (1)(4), (2)(5) and (3)(6) (see Fig. 5 (center)), that means that we must concatenate (from the right) to A0 three new columns numbered respectively 5, 6, 7, whereas the number of new lines v A0 = 0.
So, here dim 1(A1) = A1 = A0 = 6, dim2(A1) = P0 + vP0 = 4 + 3 = 7 and the matrix of the next configuration n1 (see Fig. 5 (right)) is
0 0 1 1 1 0 0
0 1 0 1 0 1 0
0 1 1 0 0 0 1
1 1 0 0 1 0 0
1 0 1 0 0 1 0
1 0 0 1 0 0 1
Figure 5. Initial configuration n0 = n4 (left) and two steps of the algorithm: adding new points (center) and new lines (right)
Note that positions of "1"-s in the concatenated columns are exactly 1 and 4, 2 and 5, and 3 and 6.
Going over to the next step s = 2 we find that
' 3 1111 3 111 13 11 113 1 1113 11113
¿1 AT =
ATA1 =
3111111 311111 31111 13111 112 0 0 110 2 0 1 1 0 0 2.
So, here dim1(A2) = A + 2 = Ai + vAi = 6 + 3 = 9, dim2(A2) = P2 = Pi + vPi = 7 + 0 = 7 and the matrix of the next configuration (see Fig. 5 (center)) is
¿2 =
0 0 111
0 0
00
101010 110001
1100100 1010010 1001001
0000110 0000101 0 0 0 0 0 1 1
Now it is not difficult to describe the general case for any step s > 0 : a1) If s = 1 mod 2 we add new points
vPs-1 = (number of non-incident lines at step s -1)
1 T = ^ (number of "0"-s in As-1 As-1)
As-1 2
As-i 2
- Total
- Total
diag ( AIS-1 As-1) 2
As-1 As-1 2
(8)
whereas clearly vAs-1 = 0.
Proof of (8): The first and second equalities are evident.
Furthermore, Total
diag ( Aj-1 As-1) 2
Total
(AS-12As-1) because (1) = 0. At last, (A,-1)
is
(diag (AT-1 As-1)
is equal to
equal to the all pairs of different lines at step s - 1, whereas Total (d 2 such pairs of lines which are already incident at this step (see item c of general properties of the incident matrices, p. 2).
For example, for s = 1 we get vP0 = (6) -4 ■ (2) = 3, since A0 = 6, diag(ATA0) = (3333).
In other words, here we get the As x Ps-matrix As, where As = As-1, Ps = Ps-1 + vPs-1, by concatenating from the right to As-1 one by one new vPs-1 columns.
So, in this case we get a formula (we remind that 0 • a = {}):
As = As-1| U ((1 - [As-1 AT-1] i,j) • m,j)
2s i S As-1 i S j S As-1
(9)
Dually,
a2) If s = 0 mod 2 we add new lines
vAs-i = (number of non-incident points at step s -1
1 T = ^ (number of "0"-s in As-1 As-1)
Ps-1 2
Ps-1 2
- Total Total
diag ( As-1 A;-1) 2
As-1 AT-1 2
(10)
whereas clearly vPs-1 = 0.
For example, for s = 2 we get vA1 = (2) -6■ (3), since P1 = 7, diag(A1 AT) = (3 333 33). So, in this case we get a formula:
As = As-1 U f(1 - [AT-1 As-1];,j) • nTj
2si S As-1 i S j s As-1
(11)
Formulas (9) and (11) give rise to the first variant of our algorithms. First four steps are illustrated on Fig. 6.
Figure 6. Incident matrix of initial configuration (a) and four first steps of the algorithm: three new points added (b), three new lines added (c), 6 new points added (d), 24 new lines added (e)
5. BILINEAR FORMS APPROACH
Let n = {pi1 and A = l1 be two sets of independent variables for points and lines respectively.
For any step s > 0 we introduce a bilinear form Fs = Fs(n, A) = nTAsA where As is an incident matrix constructed on step s (see Sec. 3) and n and A are initial segments of the infinite sequences of variables n and A having appropriate lengths. For example, for s = 0 we have n = {pi}4=1, A = {li}6=1 and
F0 = I4 p 1 + I5 p 1 + l6 P1 + l2 P2 + l3 P2 + l4 P2 + l1 P3 + l3 P3 + l5 P3 + l1 P4
+ l2 p 4 + i 6 p4 (12)
= l1(p3 + p4) + l2(p2 + p4) + l3 (p 2 + p 3) + k(p 1 + p 2) + fe(p1 + p3)
+ l6(p1 + p4)
= p 1 (¿4 + l5 + l6) + p2(l2 + l3 + l4) + p 3 (l1 + 13 + 15) + p4(l1 + l2 + fe)-
dF
Now it is clear that also in general case Coefficient [Fs, li ] = -gj1 is a linear form in n repre-
dF
senting the i-th row of As; Coefficient [Fs, pj ] = dpj is a linear form in A representing the j-th column of As.
Also it is clear that two lines, li and lk with 1 < i, k < As, i + k are not incident iff. the
dFs dlL dli and dlk
linear forms and drr have no variables in common that implies that in this case the Laplace
operator in n
and otherwise
i dFs dFs d2 t dFs dFs, _
' dlk I = ' oTk I"0 (13)
i dFs dFs
M«1=2 (14)
It's clear that if i = k then
11 —-] 1= 2 • (number of points on i - thline) = 2 [diag ( AsAl )] •
dli
(15)
For example,
whereas
and
dFo dFo
d2
■ dk) — £drf^ + P4)(P1 + ^ = 0
■ f) = £ + P4)(P2 + M = 2,
dF0\2 A d
=£ +w) >=2'2=4
Obviously that formulas dual to (13), (14) and (15) also are valid mutatis mutandis. Using formulas (13), (14), (15) and their duals it is easy to verify matrices equalities
1
(dFs
1
2MU1J = ^MUt
I dFs
— ATsAs,
(16)
dF dF
where -gj = grad\(fs), -¿n = gradn(fs), the Laplace operators are supposed to be listable and ®2 means tensor square.
Now we are going to write the recurrent formulas from step s - 1 to step s: Formulas (8) and (10) may be written in terms of bilinear forms as follows:
vPs-1 — (As-1 - Total Binomial
2
/
vAs-i — fPs-1) - Total Binomial
2 V
1 A, ( 'F-! f.2
2 dX
1 ( dFs_ 1 Ï2
As to a recurrent relation between forms Fs-1 and Fs here we have
Fs = Fs-1 + vFs-1, s > 1.
(17)
(18)
(19)
For brevity of writing formulas for we use the reduced Laplace matrices An(Fs) = J - J *
dF
dr] . For example, if s = 0 then
1A n (( f f) and AxF ) — J - J •« 1A X (( t )").
and
2a 4%
A„F (x°) —
2 1 1 ° 1 1
1 2 1 1 ° 1
1 1 2 1 1 0
— ° 1 1 2 1 1
1 ° 1 1 2 1
1 1 ° 1 1 2
° ° ° 1 ° 0 ^
° ° ° ° 1 °
° ° ° ° ° 1
1 ° ° ° ° 0
° 1 ° ° ° 0
° ° 1 ° ° °
o
i.e. in AnF(x0) all non-zero elements become "0"-s and all zeros become "1"-s. Then it is not difficult to check that for odd step, s = 1 mod 2,
vFs-i = X di + lj)P°Uj) • •' where a(i, j) = Ps-i + £ (20)
1—i <As-i,i — j sAs-i 1'j a—i ,p< j
for even step, s = 0 mod 2, s > 0
vFs-1 = X (Pi + Pj)l*(i'j) Aa . ., where a(i, j) = As-i + X LAJi,j. (21)
1 — i—Ps-1'i—j—Ps-1 1'j a— i'P—j
For example, if s = 1 then vF0 = (l1 + l4)p5 + (l2 + l5)p6 + (l5 + k)P7 and F1 = F0 + vF0 = l1(p3 + P4) + l2(p2 + P4) + l3(p2 + P3) + l4(p 1 + P2) + l5(p1 + P4) + (([1 + WP5 + (l2 + l5)p6 + (fe + l6)p7).
Formulas (19), (20), (21) are exact analogs of those (9) and (11) but the "exotic" oncatenations of matrices are changed by usual polynomial additions.
These formulas also give rise to alternative algorithm for recursive construction of fr (n4).
6. IMPLEMENTATION
As was said above we used matrix and bilinear forms approaches.
The first difficulty in programming was caused by the requirement to avoid zero-columns/rows in incident matrices as well as "fictitious" variables in bilinear forms. This difficulty is surmounted with special procedures for numeration of new constructed columns/rows of matrices and new variables of bilinear forms.
A more serious obstacle is the (above-mentioned) fact of the very fast growth of matrices' formats. Though those are very sparse 0-1-matrices, the programming tools for such matrices provided by Mathematica proved insufficient for our purposes, so computer memory resources became exhausted quickly.
As a result, we managed to calculate only 7 members of the sequence un = vPn + vAnt n > 0 (note that one of the two summand in "un" is always equal to 0):
3, 3, 6, 24, 282, 37233, 684792168,....
It is easy to check empirically that this sequence grows asymptotically as a double exponent of n (Fig. 7).
Figure 7. Number of elements grows as double exponent (linear on log(log) scale
Though we failed so far to find a general formula for the number of new elements on each step, we can find the (rather rough) upper bound using (4.1) and (4.3): ignoring second terms we immediately get
vPs-i <
As-i 2
vAs-i <
Ps-1 2
Компьютерное моделирование конечно-порожденных свободных проективных плоскостей Assuming vPs-1 = (Л2-1) and vAs_i = (Ps2-1) we have
As = As-1, Ps = Ps-1 + vPs-1 for s even,
and
Ps = Ps-1, As = As-1 + vAs-1 for s odd. See upper line on Fig. 8. We can improve this upper bound by taking into account that all diagonal elements in (4.1) and (4.3) always are > 3. We have
vPs-1 <
Л*-1 2
- 3Ps_i, vЛs-l <
Ps-1 2
3Л
s-1 •
In both cases we get double exponential growth. These two lines together with our result are shown in Fig. 8.
Figure 8. Number of elements (lower line) and two upper bounds
References
1. W.W. Sawyer, Prelude to Mathematics, Penguin Books, 1957.
2. "Combinatorial design" in Wikipedia [online]; https://en.wikipedia.org/wiki/Combinatorial_design
3. M. Hall, "Protective planes", Trans. Amer. Math. Soc., no. 54, pp. 229-277,1943.
4. M. Hall, The Theory of Groups, NY, 1959.
5. L. I. Kopejkina, "Decomposition of Protective Planes", Bull. Acad. Sci. USSR Ser. Math. Izvestia Akad. NaukSSSR, no. 9, pp. 495-526,1945.
6. R. Hartshorne, Protective Planes. Lecture Notes Harvard University, NY, 1967.
7. L. C. Siebenmann, "A Characterization of Free Projective Planes", Pac.J. of Math., vol. 15, no. 1, pp. 293298,1965.
8. R. Sandler, "The Collineation Groups of Free Planes", Trans. Amer. Math. Soc., no. 107, pp. 129-139, 1963.
9. Hang Kim Ki, F. W. Roush, "A Universal Algebra Approach to Free Projective Planes", Aequationes Mathematicae University of Waterloo, no. 18, pp. 389-400,1978.
10. A. I. Shirshov and A. A. Nikitin, Algrbraic Theory of Projective Planes, Novosibirsk, USSR: 1987.
11. H. J. Ryser, Combinatorial Mathematics, The Mathematical Association of America, 1963.
12. F. Karteszi, Introduction to Finite Geometries, Budapest, Hungary: Akademia Klado, 1976.
13. M. M. Postnikov, Lectures in Geometry, Semester 1: Analytic Geometry, Moscow, USSR: Mir Publishers, 1982.
14. M. Berger, Geometry I, Springer, 2009.
15. E. Casas-Alvero, Analytic Projective Geometry, European Mathematical Society, 2014.
Поступила в редакцию 04.07.2017, окончательный вариант — 28.07.2017.
Компьютерные инструменты в образовании, 2017 № 4:14-28
УДК: 512.56 + 514.146 http://ipo.spb.ru/journal
КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕ КОНЕЧНО-ПОРОЖДЕННЫХ СВОБОДНЫХ ПРОЕКТИВНЫХ ПЛОСКОСТЕЙ
Гогин Н. Д., Милляри А. А.1
''Университет Св. Георгия, Гренада, Вест-Индия
Аннотация
Работа посвящена компьютерному моделированию процесса построения свободных проективных плоскостей, или более точно, алгоритмическому нахождению их последовательных матриц инцидентности. Рассматриваются также некоторые целочисленные характеристики этих матриц. Матричный метод, а также подход, использующий билинейные формы, применяются для изучения темпов роста числа новых элементов (точек, линий) в процессе поэтапного построения проективной плоскости, начиная с конфигурации М. Холла П4. Число новых элементов растет асимптотически как двойная экспонента (линейно по log(log) шкале.) Оценка сверху также дает двойной экспоненциальный рост.
Ключевые слова: свободные проективные плоскости, конечные геометрии, комбинаторные схемы.
Цитирование: Gogin N. D., Myllari A. A. On Computer Modeling of Finite-generated Free Projective Planes // Компьютерные инструменты в образовании. 2017. № 4. С. 14-28.
Received 04.07.2017, the final version — 28.07.2017.
Гогин Никита Дмитриевич, кандидат физико-математических наук, доцент, [email protected]
Мюлляри Александр Альбертович, кандидат физико-математических наук, профессор, Школа Искусств и Наук, университет Сент Джорджес, Гренада, Вест-Индия, [email protected]
Nikita D. Gogin, PhD, docent, [email protected]
Aleksandr A. Mylläri,
PhD, Professor, Department of Computers and Technology, School of Arts and Sciences, St. George's University, Grenada, West Indies, [email protected]
© Our authors, 2017. Наши авторы, 2017.
