Even Astral Configurations

Abstract

A configuration $(p_q, n_k)$ is a collection of $p$ points and $n$ straight lines in the Euclidean plane so that every point has $q$ straight lines passing through it and every line has $k$ points lying on it. A configuration is astral if it has precisely $\lfloor {q+1\over2} \rfloor$ symmetry classes (transitivity classes) of lines and $\lfloor{k+1\over2} \rfloor$ symmetry classes of points. An even astral configuration is an astral configuration configuration where $q$ and $k$ are both even. This paper completes the classification of all even astral configurations.

Constructing $5$-Configurations with Chiral Symmetry

“…In [8,12], it was shown that there are two infinite families of 2-astral configurations, of the form 6k#(3k − j, 2k; j, 3k − 2j) and 6k#(2k, j; 3k − 2j, 3k − j),for k 2, 1 j < 3k/2, with j = k, along with 27 sporadic configurations in the case when m = 30, 42, 60 (plus their disconnected multiples). These configurations have been discussed in detail in other places (e.g., [7,8,10,14]). Note that in some of these references, the configuration m#(a, b; c, d) is denoted as m#a b d c .…”

Movable $(n_{4})$ Configurations

“…In [5], astral (n 4 ) configurations -that is, (n 4 ) configurations with precisely two symmetry classes each of points and lines -were completely characterized. Some astral (n 4 ) configurations, called type 1 in [2,3,4], are celestial configurations. An important result from [5], originally conjectured by Branko Grünbaum in [10], was to prove the following (modified slightly to use the current notation for celestial configurations): The sporadic configurations with m = 30, 42, and 60 are listed in Table 2.…”

“…A necessary condition for an astral celestial (n 4 ) configuration to exist is that n = 12k, for some natural number k. Here, a multiple refers to taking some number of concentric copies of a configuration, rotated so that they are equally spaced. Note that in the notation of [3,4,5,9,10], a celestial astral (n 4 ) configuration was denoted as m#a b c d , which corresponds to the configuration m#(a, b; d, c) in the notation of this paper. Finally, a remark: in the first infinite family, since j can be as large as 2k − 1, it is possible for the quantity 3k − 2j to become negative, hence the need for the absolute value.…”

“…Although (n 3 ) configurations have been studied since the late 1800s, (n 4 ) configurations have been studied for a much shorter time, initially in a series of papers by Branko Grünbaum [10,11,13]. Recently, there has been a flurry of activity concerning various kinds of geometric (n 4 ) configurations (e.g., [2,3,5,6,7,8] ).…”

Some Results on Odd Astral Configurations

“…In [3] and [2], the author presented results on a particular variety of highly symmetric geometric configurations, known as astral configurations, where q and k are both even. The current work extends those results to some cases where q or k is odd and q and k are both at least 4; for an example of such a configuration, see Figure 1.…”