Cayley-Dickson Algebras and Finite Geometry

Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \leq N \leq 6$, we first observe that the multiplication table of its imaginary units $e_a$, $1 \leq a \leq 2^N -1$, is encoded in the properties of the projective space PG$(N-1,2)$ if one regards these imaginary units as points and distinguished triads of them $\{e_a, e_b, e_c\}$, $1 \leq a<b<c \leq 2^N -1$ and $e_ae_b = \pm e_c$, as lines. This projective space is seen to feature two distinct kinds of lines according as $a+b = c$ or $a+b \neq c$. Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG$(N-1,2)$, the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a binomial $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration ${\cal C}_N$; in particular, ${\cal C}_3$ (octonions) is isomorphic to the Pasch $(6_2,4_3)$-configuration, ${\cal C}_4$ (sedenions) is the famous Desargues $(10_3)$-configuration, ${\cal C}_5$ (32-nions) coincides with the Cayley-Salmon $(15_4,20_3)$-configuration found in the well-known Pascal mystic hexagram and ${\cal C}_6$ (64-nions) is identical with a particular $(21_5,35_3)$-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. We also draw attention to a remarkable nesting pattern formed by these configurations, where ${\cal C}_{N-1}$ occurs as a geometric hyperplane of ${\cal C}_N$. Finally, a brief examination of the structure of generic ${\cal C}_N$ leads to a conjecture that ${\cal C}_N$ is isomorphic to a combinatorial Grassmannian of type $G_2(N+1)$.


Introduction
As it is well known (see, e. g., [1,2]), the Cayley-Dickson algebras represent a nested sequence A 0 , A 1 , A 2 , . . . , A N , . . . of 2 N -dimensional (in general non-associative) R-algebras with A N ⊂ A N +1 , where A 0 = R and where for any N ≥ 0 A N +1 comprises all ordered pairs of elements from A N with conjugation defined by (x, y) * = (x * , −y) (1) and multiplication usually by Every finite-dimensional algebra (see, e. g., [2]) is basically defined by the multiplication rule of its basis. The basis elements (or units) e 0 , e 1 , e 2 , . . . , e 2 N +1 −1 of A N +1 , e 0 being the real basis element (identity), can be chosen in various ways. Our preference is the canonical basis e 0 = (e 0 , 0), e 1 = (e 1 , 0), e 2 = (e 2 , 0), . . . , e 2 N −1 = (e 2 N −1 , 0), e 2 N = (0, e 0 ), e 2 N +1 = (0, e 1 ), e 2 N +2 = (0, e 2 ), . . . , e 2 N +1 −1 = (0, e 2 N −1 ), where, by abuse of notation (see, e. g., [3]), the same symbols are also used for the basis elements of A N . This is because the paper essentially focuses on multiplication properties of basis elements and the canonical basis seems to display most naturally the inherent symmetry of this operation. For, in addition to revealing the nature of the Cayley-Dickson recursive process, it also implies that for both a and b being non-zero we have e a e b = ±e a⊕b , where the symbol '⊕' denotes 'exclusive or' of the binary representations of a and b (see, e. g., [4]). From the above expressions and Eqs. (1) and (2) one can readily find the product of any two distinct units of A N +1 if the multiplication properties of those of A N are given. Such products are usually expressed/presented in a tabular form, and we shall also follow this tradition here. All the multiplication tables we made use of were computed for us by Jörg Arndt; the computer program is named cayley-dickson-demo.cc and is freely available from his web-site http://jjj.de/fxt/demo/arith/index.html.
Employing such a multiplication table of A N , N ≥ 2, it can be verified that the 2 N − 1 imaginaries e a , 1 ≤ a ≤ 2 N − 1, form 2 N −1 2 /3 distinguished sets each of which comprises three different units {e a , e b , e c } that satisfy equation and where each unit is found to belong to 2 N −1 − 1 such sets. 1 Regarding the imaginaries as points and their distinguished triples as lines, one gets a point-line incidence geometry where every line has three points and through each point there pass 2 N −1 −1 lines and which is isomorphic to PG(N − 1, 2), the (N − 1)-dimensional projective space over the smallest Galois field GF (2) (see also [5] for N = 3 and [6] for N = 4). Let us assume, without loss of generality, that the elements in any distinguished triple {e a , e b , e c } of A N are ordered in such a way that a < b < c. Then, for N ≥ 3, we can naturally speak about two different kinds of triples and, hence, two distinct kinds of lines of the associated 2 N -nionic PG(N − 1, 2), according as a + b = c or a + b = c; in what follows a line of the former/latter kind will be called ordinary/defective. This stratification of the line-set of the PG(N − 1, 2) induces a similar partition of the point-set of the latter space into several types, where a point of a given type is characterized by the same number of lines of either kind that pass through it. Obviously, if our projective space PG(N − 1, 2) is regarded as an abstract geometry per se, every point and/or every line in it has the same footing. So, to account for the abovedescribed 'refinement' of the structure of our 2 N -nionic PG(N − 1, 2), it turns out to be necessary to find a representation of this space where each point/line is ascribed a certain 'internal' structure, which at first sight may seem to be quite a challenging task. To tackle this task successfully, we need to introduce a few notions/concepts from the realm of finite geometry.
We start with a finite point-line incidence structure C = (P, L, I) where P and L are, respectively, finite sets of points and lines and where incidence I ⊆ P × L is a binary relation indicating which point-line pairs are incident (see, e. g., [7]). Here, we shall only be concerned with specific point-line incidence structures called configurations [8]. A (v r , b k )-configuration is a C where: 1) v = |P| and b = |L|, 2) every line has k points and every point is on r lines, and 3) two distinct lines intersect in at most one point and every two distinct points are joined by at most one line; a configuration where v = b and r = k is called symmetric (or balanced), and usually denoted as a (v r )-configuration. A (v r , b k )-configuration with v = r+k−1 r and b = r+k−1 k is called a binomial configuration. Next it comes a geometric hyperplane of C = (P, L, I), which is a proper subset of P such that a line from C either lies fully in the subset, or shares with it only one point. If C possesses geometric hyperplanes, then one can define the Veldkamp space of C, V(C), as follows [9]: (i) a point of V(C) is a geometric hyperplane of C and (ii) a line of V(C) is the collection where H ′ and H ′′ are distinct geometric hyperplanes. If each line of C has three points and C 'behaves well,' a line of V(C) is also of size three and can equivalently be defined as {H ′ , H ′′ , H ′ ∆H ′′ }, where the symbol ∆ stands for the symmetric difference of the two geometric hyperplanes and an overbar denotes the complement of the object indicated. From its definition it is obvious that V(C) is well suitable for our needs because its points, being themselves sets of points, have different 'internal' structure and so, in general, they can no longer be on the same par; clearly, the same applies to the lines V(C). Our task thus basically boils down to finding such C N whose V(C N ) is isomorphic to PG(N − 1, 2) and completely reproduces its 2 N -nionic fine structure. This will be carried out in great detail for the first four non-trivial cases, 3 ≤ N ≤ 6, which, when combined with the two trivial cases (N = 1, 2), will provide us with sufficient amount of information to guess a general pattern.
The paper is organized as follows. In Sect. 2 it is shown that C 3 (octonions) is isomorphic to the Pasch (6 2 , 4 3 )-configuration, which plays a key role in classifying Steiner triple systems. In Sect. 3 one demonstrates that C 4 (sedenions) is nothing but the famous Desargues (10 3 )-configuration. In Sect. 4 our C 5 (32-nions) is shown to be identical with the Cayley-Salmon (15 4 , 20 3 )-configuration found in the well-known Pascal mystic hexagram. In Sect. 5 we find that C 6 corresponds to a particular (21 5 , 35 3 )-configuration encompassing seven distinct copies of the Cayley-Salmon (15 4 , 20 3 )-configuration as geometric hyperplanes. In Sect. 6 some rudimentary properties of the generic C N ∼ = N +1 2 N −1 , N +1 3 3 -configuration are outlined and its isomorphism to a combinatorial Grassmannian of type G 2 (N + 1) is conjectured. Finally, Sect. 7 is reserved for concluding remarks.
2 Octonions and the Pasch (6 2 , 4 3 )-configuration From the nesting property of the Cayley-Dickson construction of A N it is obvious that the smallest non-trivial case to be addressed is A 3 , the algebra of octonions, whose multiplication table is presented in Table 1. Table 1: The multiplication table of the imaginary unit octonions e a , 1 ≤ a ≤ 7. For the sake of simplicity, in what follows we shall employ a short-hand notation e a ≡ a; likewise for the real unit e 0 ≡ 0. There are also delineated multiplication tables corresponding to the distinguished nested sequence of sub-algebras of complex numbers (a = 1, the upper left corner) and quaternions (1 ≤ a ≤ 3, the upper left 3 × 3 square). * 1 2 3 The above-given multiplication table implies the existence of the following seven distinguished trios of imaginary units: Regarding the seven imaginary units as points and the seven distinguished triples of them as lines, we obtain a point-line incidence structure where each line has three points and, dually, each point is on three lines, and which is isomorphic to the smallest projective plane PG(2, 2), often called the Fano plane, depicted in Figure 1. It is then readily seen that we have six ordinary lines, namely Similarly, our octonionic PG(2, 2) features two distinct types of points. A type-one point is such that two lines passing through it are ordinary, the remaining one being defective; such a point lies in the set {3, 5, 6} ≡ α.
A type-two point is such that every line passing through it is ordinary; such a point belongs to the set {1, 2, 4, 7} ≡ β, which is highlighted by gray color in Figure 1. A configuration C 3 whose Veldkamp space reproduces the above-described partitions of points and lines of PG(2, 2) is, as we will soon see, nothing but the well-known Pasch (6 2 , 4 3 )configuration, P. This configuration, which plays a very important role in classifying Steiner triple systems (see, e. g., [10]), is depicted in Figure 2 in a form showing an automorphism of order three; it also lives in the Fano plane and, as it is readily seen by comparing Figures  1 and 2, it can be obtained from the latter by removal of any of its seven points and all the three lines passing through it.
In order to see that V(P) ∼ = PG(2, 2) we shall first show, using our diagrammatical representation of P, all seven geometric hyperplanes of P - Figure 3. We see that they are indeed of two different forms, of cardinality three and four. A member of the former set comprises two points at maximum distance from each other. Such geometric hyperplane corresponds to a type-one (or α-) point of PG (2,2). A member of the latter set features three points on a common line; such a geometric hyperplane of P corresponds to a typetwo (or β-) point of our PG(2, 2). The seven lines of V(P) are illustrated in a compact diagrammatic form in Figure 4; as it is easily discernible, each of the six ordinary lines is of the form {α, β, β}, whilst the remaining defective one has the {α, α, α} shape. Figure 2: An illustrative portrayal of the Pasch configuration: circles stand for its points, whereas its lines are represented by triples of points on common straight segments (three) and the triple lying on a big circle.

Sedenions and the Desargues (10 )-configuration
Our next focus is on A 4 , the sedenions, whose basic multiplication properties are summarized in Table 2.
An inspection of this table yields as many as 35 distinguished triples, namely: Regarding the 15 imaginary units as points and the 35 distinguished trios of them as lines, we obtain a point-line incidence structure where each line has three points and each point is on seven lines, and which is isomorphic to PG(3, 2), the smallest projective spaceas depicted in Figure 5. The latter figure employs a diagrammatical model of PG (3,2) built, after Polster [11], around the pentagonal model of the generalized quadrangle of type GQ(2, 2) whose 15 lines are illustrated by triples of points lying on black line-segments (10 of them) and/or black arcs of circles (5). The remaining 20 lines of PG(3, 2) comprise four distinct orbits: the yellow, red, blue and green one consisting, respectively, of the yellow Similarly, our sedenionic PG(3, 2) features two distinct types of points. A type-one point is such that four lines passing through it are ordinary, the remaining three being defective; such a point lies in the set {3, 5, 6, 7, 9, 10, 11, 12, 13, 14} ≡ α.
A type-two point is such that every line passing through it is ordinary; such a point belongs to the set being illustrated by gray shading in Figure 5. We see that all defective lines are of the same form, namely {α, α, α}. The 25 ordinary lines split into two distinct families. A configuration C 4 whose Veldkamp space reproduces the above-described partitions of points and lines of PG (3,2) is, as demonstrated below, the famous Desargues (10 3 )configuration, D, which is one of the most prominent point-line incidence structures (see, e. g., [12]). Up to isomorphism, there exist altogether ten (10 3 )-configurations. The Desargues configuration is, unlike the others, flag-transitive and the only one where for each of its points the three points that are not collinear with it lie on a line. This configuration, depicted in Figure 6 in a form showing its automorphism of order three, also lives in our sedenionic PG(3, 2); here, its points are the ten α-points and its lines are all the defective lines. In order to see that V(D) ∼ = PG(3, 2) we shall first introduce, using our diagrammatical representation of D, all 15 geometric hyperplanes of D - Figure 7. We see that they are indeed of two different forms, and of required cardinality ten and five. A member of the former set comprises a point and three points not collinear with it. Such geometric hyperplane corresponds to a type-one (or α-) point of PG (3,2). A member of the latter set features six points located on four lines, with two lines per each point; this is nothing but the Pasch configuration we introduced in the previous section. Such a geometric hyperplane of D corresponds to a type-two (or β-) point of our PG (3,2). It is also a straightforward task to verify that V(D) is endowed with 35 lines splitting into the required three families; those that correspond to defective lines of our sedenionic PG (3,2) are shown in Figure 8, while those that correspond to ordinary lines are depicted in Figure 9 (of type {α, β, β}) and Figure 10 (of type {α, α, β}). Figure 11 offers a 'condensed' view of the isomorphism V(D) ∼ = PG (3,2).
We shall finalize this section by pointing out that the existence of two different kinds of geometric hyperplanes of the Desargues configuration is closely connected with two wellknown views of this configuration. The first one is as a pair of triangles that are in perspective from both a point and a line (Desargues' theorem), the point and the line forming a geometric hyperplane. The other view is as the incidence sum of a complete quadrangle (i. e., a (4 3 , 6 2 )configuration) and a Pasch (6 2 , 4 3 )-configuration [17].

-configuration
Our next case is A 5 , or the 32-nions, whose multiplication properties are encoded in Table  3. From this    A point-line configuration C 5 whose Veldkamp space accounts for these stratifications of both the point-and line-set of our 32-nionic PG(4, 2) is of type (15 4 , 20 3 ). 2 This configuration is formed within our PG(4, 2) by 15 β-points and 20 defective lines of {β, β, β} type and its structure is sketched in Figure 12. It is a rather easy task to verify that this particular (15 4 , 20 3 )-configuration possesses 31 distinct geometric hyperplanes that fall into three different types. A type-one hyperplane consists of a pair of skew lines at maximum distance from each other; there are, as depicted in Figure 13, ten hyperplanes of this type and they correspond to α-points of PG (4,2). A type-two hyperplane features a point and all the points not collinear with it, the latter forming -not surprisingly -the Pasch configuration; there are, as shown in Figure 14, fifteen hyperplanes of this type and their counterparts are β-points of PG (4,2). A type-three hyperplane is identical with the Desargues configuration; we find, as portrayed in Figure 15, altogether six guys of this type, each standing for a γ-point of PG (4,2).
We also find that our (15 4 , 20 3 )-configuration yields 155 Veldkamp lines that are, as expected, of five different types. A type-I Veldkamp line, shown in Figure 16a, features two hyperplanes of type one and a type-two hyperplane and its core consists of two points that are at maximum distance from each other; there are 10 2 = 15 × 6/2 = 45 Veldkamp lines of this type and they correspond to defective lines of PG(4, 2) of type {α, α, β}. A type-II Veldkamp line, featured in Figure 16b, is composed of three hyperplanes of type two that share three points on a common line; there are, obviously, 20 Veldkamp lines of this type, having for their counterparts defective lines of PG(4, 2) of type {β, β, β}. A type-III Veldkamp line, portrayed in Figure 16c, also consists of three hyperplanes of type two, but in this case the three common points are pairwise at maximum distance from each other; a quick count leads to 15 Veldkamp lines of this type, these being in a bijection with 15 ordinary lines of PG(4, 2) of type {β, β, β}. Next, it comes a type-IV Veldkamp line, depicted in Figure 16d, which exhibits a hyperplane of each type and whose core is composed of a  Figure 6. The five points added to the Desargues configuration are the three peripheral points and the red and blue point in the center. The ten lines added are three lines denoted by red color, three blue lines, three lines joining pairwise the three peripheral points and the line that comprises the three points in the center of the figure, that is the ones represented by a bigger red circle, a smaller blue circle and a medium-sized black one. 1 3 3 3 Figure 13: The ten geometric hyperplanes of the (15 4 , 20 3 )-configuration of type one; the number below a subfigure indicates how many hyperplane's copies we get by rotating the particular subfigure through 120 degrees around its center.    Figures 8 to 10, each representative of a geometric hyperplane is drawn separately and different colors are used to distinguish between different hyperplane types: red is reserved for type one, yellow for type two and blue for type three hyperplanes. As before, black color denotes the core of a Veldkamp line, that is the elements common to all the three hyperplanes comprising it. line and a point at the maximum distance from it; since for each line of our (15 4 , 20 3 )configuration there are three points at maximum distance from it, there are 20 × 3 = 60 Veldkamp lines of this type, having their twins in ordinary lines of PG(4, 2) of type {α, β, γ}. Finally, we meet a type-V Veldkamp line, sketched in Figure 16e, which is endowed with two hyperplanes of type three and a single one of type two, and whose core is isomorphic to the Pasch configuration; hence, we have 6 2 = 15 Veldkamp lines of this type, being all representatives of ordinary lines of PG(4, 2) of type {β, γ, γ}.
Before embarking on the final case to be dealt with in detail, it is worth having a closer look at our (15 4 , 20 3 )-configuration and pointing out its intimate relation with famous Pascal's Mystic Hexagram. If six arbitrary points are chosen on a conic section and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet in three points that lie on a straight line, the latter being called the Pascal line. Taking the permutations of the six points, one obtains 60 different hexagons. Thus, the so-called complete Pascal hexagon determines altogether 60 Pascal lines, which generate a remarkable configuration of 146 points and 110 lines called the hexagrammum mysticum, or the complete Pascal figure (for the most comprehensive, applet-based representation of this remarkable geometrical object, see [14]). Both the points and lines of the complete Pascal figure split into several distinct families, usually named after their discoverers in the first half of the 19th century. We are concerned here with the 15 Salmon points and the 20 Cayley lines (see, e. g. [15,16]) which form a (15 4 , 20 3 )-configuration. This configuration is discussed in some detail in [17], where it is also depicted ( Figure 6) and called the Cayley-Salmon (15 4 , 20 3 )configuration. And it is precisely this Cayley-Salmon (15 4 , 20 3 )-configuration which our 32-nionic (15 4 , 20 3 )-configuration is isomorphic to. The same configuration is also portrayed in Figure 8 of [14]. In the latter work, two different views/interpretations of the configuration are also mentioned. One is as three pairwise-disjoint triangles that are in perspective from a line, in which case the centers of perspectivity are guaranteed by Desargues' theorem to also lie on a line; we just stress here that these two lines form a geometric hyperplane (of type one, see Figure 13). The other view of the figure takes any point of the configuration to be the center of perspectivity of two quadrangles whose six pairs of corresponding sides meet necessarily in the points of a Pasch configuration; again, the point and the associated Pasch configuration form a geometric hyperplane (of type two, see Figure 14). Obviously, we can offer one more view of the configuration, that stemming from the existence of type-three hyperplanes, namely as the incidence sum of a Desargues configuration and three triangles on a commmon side (see Figure 15).

64-nions and a (21 , 33 )-configuration
The final algebra we shall treat in sufficient detail is A 6 , or the 64-nions. From the corresponding multiplication table, which due to its size we do not show here but which is freely available at http://jjj.de/tmp-zero-divisors/mult-table-64-ions.txt, we infer the existence of 651 distinguished triples of imaginary units. Regarding the 63 imaginary units of 64-nions as points and the 651 distinguished triples of them as lines, we obtain a point-line incidence structure where each line has three points and each point is on 31 lines, and which is isomorphic to PG (5,2). Following the usual procedure, we find that 350 lines of this space are defective and 301 ordinary. Likewise the preceding case, we encounter three different types of points in our 64-nionic PG  The Veldkamp space mimicking such a fine structure of PG(5, 2) is that of a particular (21 5 , 35 3 )-configuration, C 6 , that also lives in our PG (5,2) and whose points are the 21 β-points and whose lines are the 35 defective lines of {β, β, β} type. To visualise this configuration, we build it around the model of the Cayley-Salmon (15 4 , 20 3 )-configuration of 32-nions shown in Figure 12. Given the Cayley-Salmon configuration, there are six points and 15 lines to be added to yield our (21 5 , 35 3 )-configuration, and this is to be done in such a way that the configuration we started with forms a geometric hyperplane in it. As putting all the lines into a single figure would make the latter look rather messy, in Figure 17 we briefly illustrate this construction by drawing six different figures, each featuring all six additional points (gray) but only five out of 15 additional lines (these lines being also drawn in gray color), namely those passing through a selected additional point (represented by a doubled circle). Employing this handy diagrammatical representation, one can verify that our (21 5 , 35 3 )-configuration exhibits 63 geometric hyperplanes that fall into three distinct types. A type-one hyperplane consists of a line and its complement, which is the Pasch configuration; there are 35 distinct hyperplanes of this form, each corresponding to an αpoint of our PG (5,2). A type-two hyperplane comprises a point and its complement, which is the Desargues configuration; there are 21 hyperplanes of this form, each having a β-point for its PG(5, 2) counterpart. Finally, a type-three hyperplane is isomorphic to the Cayley-Salmon configuration; there are seven distinct guys of this type, each answering to a γ-point of the PG(5, 2). We leave it with the interested reader to verify by themselves that the Veldkamp space of our (21 5 , 35 3 )-configuration indeed features 651 lines that do fall into the above-mentioned seven distinct kinds.
As in the previous two cases, we shall briefly describe a couple of interesting views of our (21 5 , 35 3 )-configuration, both related to type-one hyperplanes. The first one is as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration, the line and the Pasch configuration comprising a geometric hyperplane (compare with the first view of both the Desargues and the Cayley-Salmon configuration). This is sketched in Figure 18, where the four triangles are denoted, in boldfacing, by green, red, yellow and blue color, the line of perspectivity by boldfaced gray color, and the points of perspectivity of pairs of triangles (together with the corresponding lines they lie on and that are also boldfaced) by black color. The other view is as three complete quadrangles that are pairwise in perspective in such a way that the three points of perspectivity lie on a line and where the six triples of their corresponding sides meet at points located on a Pasch configuration, again the line and the Pasch configuration forming a geometric hyperplane (compare with the second view of both the Desargues and the Cayley-Salmon configuration). It is curious to notice that the first entry represents a triangular number, while the second one is a tetrahedral number, or triangular pyramidal number. In other words, we get a nested  sequence of binomial ( r+k−1 r r , r+k−1 k k )-configurations with r = N − 1 and k = 3, whose properties have very recently been discussed in a couple of interesting papers [18,19]. The first few configurations are shown, in a form where the configurations are nested inside each other, in Figure 19.
A particular character of this nesting is reflected in the structure of geometric hyperplanes. Denoting our generic N +1 2 N −1 , N +1 3 3 -configuration by C N , we can express the types of geometric hyperplanes of the above-discussed cases in a compact form as follows which implies the following generic hyperplane compositions according as N is even or odd, respectively; here, the symbol '⊔' stands for a disjoint union of two sets.
In the spirit of previous sections, let us also have a closer look at the nature of our generic C N . To this end, we first recall the following observations. C 4 , the Desargues configuration, can be viewed as (4 − 2 =) two triangles in perspective from a line which are also perspective from a point, that is C 1 ; the line and the point form a geometric hyperplane of C 4 . Next, C 5 , the Cayley-Salmon configuration, admits a view as (5 − 2 =) three triangles in perspective from a line where the points of perspectivity of three pairs of them are on a line, aka C 2 ; the two lines form a geometric hyperplane of C 5 . Finally, C 6 , our (21 5 , 35 3 )-configuration, can be treated as (6 − 2 =) four triangles in perspective from a line where the points of perspectivity of six pairs of them lie on a Pasch configuration, alias C 3 ; the line and the Pasch configuration form a geometric hyperplane of C 6 . Generalizing these observations, we conjecture that for any N ≥ 4, C N can be regarded as N − 2 triangles that are in perspective from a line in such a way that the points of perspectivity of N −2 2 pairs of them form the configuration isomorphic to C N −3 , where the latter and the axis of perspectivity form a geometric hyperplane of C N .
Next, we invoke the concept of combinatorial Grassmannian (see, e. g., [20,21]). Briefly, a combinatorial Grassmannian G k (|X|), where k is a positive integer and X is a finite set, is a point-line incidence structure whose points are k-element subsets of X and whose lines are (k + 1)-element subsets of X, incidence being inclusion. It is known [20] that if |X| = N + 1 and k = 2, G 2 (N + 1) is a binomial N +1 2 N −1 , N +1 3 3 -configuration; in particular, G 2 (4) is the Pasch configuration, G 2 (5) is the Desargues configuration and G 2 (N +1)'s with N ≥ 5 are called generalized Desargues configurations. Now, from our detailed examination of the four cases it follows that C 3 , C 4 , C 5 , C 6 , . . . , C N are endowed with 1, 5, 15, 35, . . . , N +1 4 Pasch configurations. And as N +1 4 is also the number of Pasch configurations in G 2 (N +1), N ≥ 3, we are also naturally led to conjecture that C N ∼ = G 2 (N + 1). From what we have found in the previous sections it follows that this property indeed holds for 1 ≤ N ≤ 6, being illustrated for N = 5 and N = 6 in Figure 20. Right: -A pictorial illustration of C 6 ∼ = G 2 (7). Here, the labels of six additional points are only depicted, the rest of the labeling being identical to that shown in the left-hand side figure.

Conclusion
An intriguing finite-geometrical underpinning of the multiplication tables of Cayley-Dickson algebras A N , 3 ≤ N ≤ 6, has been found that admits generalization to any higherdimensional A N . This started with an observation that the multiplication properties of imaginary units of the algebra A N are encoded in the structure of the projective space PG(N − 1, 2). Next, this space was shown to possess a refined structure stemming from particular properties of triples of imaginary units forming its lines. To account for this refinement, we employed the concept of Veldkamp space of point-line incidence structure and found out the latter to be a binomial N +1 2 N −1 , N +1 3 3 -configuration C N ; in particular, C 3 (octonions) was found to be isomorphic to the Pasch (6 2 , 4 3 )-configuration, C 4 (sedenions) to the famous Desargues (10 3 )-configuration, C 5 (32-nions) to the Cayley-Salmon (15 4 , 20 3 )configuration found in the well-known Pascal mystic hexagram and C 6 (64-nions) was shown to be identical with a particular (21 5 , 35 3 )-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. These configurations are seen to form a remarkable nested pattern, where C N −1 is embedded in C N as its geometric hyperplane, that naturally reflects the spirit of the Cayley-Dickson recursive construction of corresponding algebras.
It is a well-known fact that the only first four algebras A N , 0 ≤ N ≤ 3, are 'wellbehaving' in the sense of being normed, alternative and devoid of zero-divisors -the facts that are frequently offered as an explanation why a relatively little attention has been paid so far to their higher-dimensional cousins, these latter being even regarded by some scholars as 'pathological.' It may well be that our finite-geometric, Veldkamp-space-based approach will be able to shed a novel, unexpected light at this issue as it is only starting with N = 4 when C N is found to feature a 'generalized Desargues property' in the sense that it can be interpreted as N −2 triangles that are in perspective from a line in such a way that the points of perspectivity of N −2 2 pairs of them form the configuration isomorphic to C N −3 . Or, in a slightly different form, it is only for N ≥ 4 when C N contains Desargues configurations, these occurring as components of its geometric hyperplanes at that.