Eventually, geometric $(n_{k})$ configurations
exist for all $n$

Starting from an $(m_{k-1})$ configuration $\mathcal{C}$ we construct a new configuration $\mathcal{D}$ which is a $((k+1)m_{k})$ configuration. A sketch of the construction is that $k-1$ affine images of $\mathcal{C}$ are carefully constructed so that each point $P$ of $\mathcal{C}$ is collinear with the $k-1$ images of $P$, and each line of $\mathcal{C}$ and its images are concurrent at a single point. Then $\mathcal{D}$ consists of the the points and lines of $\mathcal{C}$ and its images, the new lines corresponding to the collinearities from each point $P$, and the new points of concurrence corresponding to the lines of $\mathcal{C}$ and their images.

The details of the construction are as follows:

1. (1)

   Let $A$ be a line that (i) does not pass through the intersection of two lines of $\mathcal{C}$, whether or not that intersection point is a point of the configuration; (ii) is perpendicular to no line connecting any two points of $\mathcal{C}$, whether or not that line is a line of the configuration; (iii) intersects all lines of $\mathcal{C}$.

2. (2)

   Let $\alpha_{1},\alpha_{2},\ldots\alpha_{k-1}$ be pairwise different orthogonal axial affinities with axis $A$. Construct copies $\mathcal{C}_{1}=\alpha_{1}(\mathcal{C})$, $\mathcal{C}_{2}=\alpha_{2}(\mathcal{C})$,…, $\mathcal{C}_{k-1}=\alpha_{k-1}(\mathcal{C})$ of $\mathcal{\mathcal{C}}=\mathcal{C}_{0}$.

3. (3)

   Let $\ell$ be any line of $\mathcal{C}$. Since $A$ is the common axis of each $\alpha_{i}$, the point $A\cap\ell$ is fixed by all these affinities. This means that the $k$-tuple of lines $\ell,\alpha_{1}(\ell),\ldots,\alpha_{k-1}(\ell)$ has a common point of intersection lying on $A$. We denote this point by $F_{\ell}$. By condition (i) in (1), for different lines $\ell,\ell^{\prime}\in\mathcal{C}$ the points $F_{\ell},F_{\ell^{\prime}}$ differ from each other; they also differ from each point of the configurations $\mathcal{C}_{i}$ $(i=1,2,\dots,k-1)$. We denote the set $\{F_{\ell}:\ell\in\mathcal{C}\}$ of points lying on $A$ by $\mathcal{F}$.

4. (4)

   Let $P$ be any point of $\mathcal{C}$. Since the affinities $\alpha_{i}$ are all orthogonal affinities (with the common axis $A$), the $k$-tuple of points $P,\alpha_{1}(P),\dots,\alpha_{k-1}(P)$ lies on a line perpendicular to $A$ (and avoids $A$, by condition (i)). We denote this line by $\ell_{P}$. Clearly, we have altogether $m$ such lines, one for each point of $\mathcal{C}$, with no two of them coinciding, by condition (ii). We denote this set $\{\ell_{P}:P\in\mathcal{C}\}$ of lines by $\mathcal{L}$.

5. (5)

   Put $\mathcal{D}=\mathcal{C}_{0}\cup\mathcal{C}_{1}\cup\dots\cup\mathcal{C}_{k-1}\cup\mathcal{F}\cup\mathcal{L}$.

We may summarize the above discussion as follows:

In our description of this construction, we are inspired by Grünbaum [11, §3.3, pp. 177–180] but we have chosen a slightly different approach (in particular, we avoid using 3-space). At the same time, we generalize it from $(m_{4})$ to $(m_{k})$.

A sketch of the construction is as follows: Suppose that $\mathcal{C}$ is an $(m_{k})$ configuration that contains a pencil $\mathcal{P}$ of $p$ parallel lines in one direction, and a pencil $\mathcal{Q}$ of $q$ parallel lines in a second direction, where the two pencils share no common configuration points; we say that the pencils are independent. For each subpencil $\mathcal{S}$ of $\mathcal{P}$ and $\mathcal{T}$ of $\mathcal{Q}$ containing $s$ parallel lines and $t$ parallel lines respectively, with $1\leq s\leq p$ and $0\leq t\leq q$, we form the subfiguration $\hat{\mathcal{C}}$ by deleting $\mathcal{S}$ and $\mathcal{T}$ from $\mathcal{C}$ (here we use the term subfiguration in the sense of Grünbaum [11]). We then carefully construct $k-2$ affine images of $\hat{\mathcal{C}}$ in such a way that for each (deleted) line $\ell$ in $\mathcal{S}$ and for each point $P_{1},P_{2},\ldots P_{k}$ on $\ell$, the collection of lines through each $P_{i}$ and its images all intersect in a single point $Y_{\ell}$, and simultaneously, for each line $\ell^{\prime}$ in $\mathcal{T}$ and for each point $Q_{1},Q_{2},\ldots Q_{k}$ on $\ell^{\prime}$, the collection of lines through $Q_{i}$ and its images all intersect in a single point $X_{\ell^{\prime}}$. Let $\mathcal{D}$ be the collection of all the undeleted points and lines of $\hat{\mathcal{C}}$ and its affine images and for each of the deleted $\ell$ and $\ell^{\prime}$, the new lines through each point $P_{i}$ $Q_{i}$ and their images, the points $Y_{\ell}$, and the points $X_{\ell^{\prime}}$; then $\mathcal{D}$ is a $(((k-1)m+s+t)_{k})$ configuration.

As a preparation, we need the following two propositions.

Now we apply the following construction. Let $\mathcal{C}$ be an $(m_{k})$ configuration such that it contains a pencil $\mathcal{P}$ of $p\geq 1$ parallel lines and a pencil $\mathcal{Q}$ of $q\geq 1$ parallel lines, too, such that these pencils are perpendicular to each other and are independent.Note that any configuration containing independent pencils in two different directions can be converted by a suitable affine transformation to a configuration in which these pencils will be perpendicular to each other.

Choose a position of $\mathcal{C}$ (applying an affine transformation if necessary) such that these pencils are parallel to the $x$-axis and $y$-axis, respectively.

1. (1)

   Remove lines $\ell_{1},\dots,\ell_{s}$ $(s\leq p)$ from the pencil $\mathcal{P}$ parallel to the $x$-axis and $\ell_{s+1},\dots,\ell_{s+t}$ $(0\leq t\leq q)$ from the pencil $\mathcal{Q}$ parallel to the $y$-axis. Let $\widehat{\mathcal{C}}$ denote the substructure of $\mathcal{C}$ obtained in this way.

2. (2)

   Let $h$ be a positive integer (say, some suitable multiple of $k$), and for each $j$, $j=1,\dots,k-2$, let $\alpha_{j}$ be an affine transformation defined in Proposition 7. Form the images $\alpha_{j}(\widehat{\mathcal{C}})$ for all $j$ given here.

3. (3)

   Let $P$ be a point of $\widehat{\mathcal{C}}$ that was incident to one of the lines $\ell_{i}$ removed from $\mathcal{C}$. Take the images $\alpha_{j}(P)$ for all $j$ given in (2). By Proposition 7, all the $k-1$ points $P,\alpha_{j}(P)$ are collinear. Let $c_{i}(P)$ denote this line.

4. (4)

   Take all the configuration points on $\ell_{i}$ and repeat (3) for each of them. By Proposition 6, the $k$-set of lines $\{c_{i}(P):P\in\ell_{i}\}$ form a pencil whose center lies on the $x$-axis or the $y$-axis according to which axis $\ell_{i}$ is perpendicular to.

5. (5)

   Let $r=(s+t)\in\{1,2,\dots,p+q\}$ be the number of lines removed from the pencils of $\mathcal{C}$ in the initial step of our construction. Repeat (4) for all these lines. Eventually, we obtain $rk$ new lines and $r$ new points such that the set of the new lines is partitioned into $r$ pencils, and the new points are precisely the centers of these pencils (hence they lie on the coordinate axes). Observe that there are precisely $k$ lines passing through each of the new points, and likewise there are precisely $k$ points lying on each of the new lines.

6. (6)

   Putting everything together, we form a $(((k-1)m+r)_{k})$ configuration, whose

   • •

     points come from the $(k-1)m$ points of the copies of $\widehat{\mathcal{C}}$, completed with the $r$ new points considered in (5).

   • •

     lines come from the $(k-1)(m-r)$ lines of the copies of $\widehat{\mathcal{C}}$, completed with the $rk$ new lines considered in (5).

   We use the notation $AS(m,k,r)$ to represent the $(((k-1)m+r)_{k})$ configuration described above.

Summarizing the discussion above, we conclude:

Note that $p+q$ independent lines in an $(m_{k})$ configuration covers $k(p+q)\leq m$ points. This gives an upper bound $p+q\leq m/k$, where the equality is attained only if $m$ divides $k$.

In this paper we use the above Lemma 8 in connection with Lemma 5 only for the case of a single pencil of parallel lines, such that $p=0$.

This collection of consecutive configurations is represented by the notation $\mathrm{AS+}(m,k,q)$. That is, $\mathrm{AS+}(m,k,q)=[AS(m,k,r)]_{r=1}^{q}.$

Since axial affinities play a crucial role in the constructions described above, we recall a basic property. The proof of the following proposition is constructive, hence it provides a simple tool for a basically synthetic approach to these constructions, which is especially useful when using dynamic geometry software to construct these configurations.

Start again with a configuration $\mathcal{C}$, and assume that the pencil $\mathcal{P}$ is in horizontal position; accordingly, the axis that we use is in vertical position (see e.g. Figure 5). Choose a line $\ell$ in $\mathcal{P}$, and a configuration point $P_{0}$ on $\ell$; then, remove $\ell$. $P_{0}$ will be the initial point of our construction (e.g., in Figure 5 the “north-west” (black) point of the starting configuration). Choose a point $C_{\ell}$ on the axis such that the line $C_{\ell}P_{0}$ is not perpendicular to the axis (in our example, this is the red point in Figure 5a).

Now let $t\in\mathbb{R}$ be our continuous parameter. Take the point

thus $P$ is a point on the line $C_{\ell}P_{0}$, and as $t$ changes, $P$ slides along this line. Moreover, by Proposition 10 we see that the pair of points $(P_{0},P)$ determines two orthogonal affinities whose axes are perpendicular to each other. In particular, the axes are precisely the coordinate axes. These affinities act simultaneously, i.e. $P_{0}$ is sent to $P$ by their (commuting) product. Using coordinates, such as $P_{0}(x_{0},y_{0})$ and $P(x,y)$, we also see that the ratio of these affinities is $y/y_{0}$ (for that with horizontal axis), respectively $x/x_{0}$ (for that with vertical axis). (Note that these ratios, using the relation (2), can also be expressed by the parameter $t$ and by the prescribed coordinates of $P_{0}$ and $C_{\ell}$. Furthermore, similarly, the matrix (1) above can also be parametrized by $t$; we omit the details.)

It is easily checked that both Proposition 6 and Proposition 7 remains valid with this continuous parameter $t$. Hence, for any $P$, we can construct the corresponding affine image of $\mathcal{C}$ (or its substructures $\hat{\mathcal{C}}$ with lines of any number $r$ removed), together with the new lines (which are denoted by red in our example of Figure 5). In particular, in case of $k$-configurations, we need to choose altogether $k-2$ points on the line $C_{\ell}P_{0}$ (note the for $t=0$, the starting copy $\mathcal{C}$ returns, for $t=1$ the image of $\mathcal{C}$ collapses to a segment within the $y$-axis, and for a third value depending on the slope of $C_{\ell}P_{0}$, it collapses to a segment within the $x$-axis; these cases thus are to be avoided).