A pentagonal geometry $\text{PENT}\left(k,r\right)$ is a partial linear space, where every line, or block, is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. An opposite line pair in a pentagonal geometry consists of two parallel lines such that each point on one of the lines is not collinear with precisely those points on the other line. We give a direct construction for an infinite sequence of pentagonal geometries with block size 3 and connected deficiency graphs. Also we present 39 new pentagonal geometries with block size 4 and five with block size 5, all with connected deficiency graphs. Consequentially we determine the existence spectrum up to a few possible exceptions for $\text{PENT}\left(4,r\right)$ that do not contain opposite line pairs and for $\text{PENT}\left(4,r\right)$ with one opposite line pair. More generally, given $j$ we show that there exists a $\text{PENT}\left(4,r\right)$ with $j$ opposite line pairs for all sufficiently large admissible $r$. Using some new group divisible designs with block size 5 (including types $2^{35},2^{71}$, and $10^{23}$) we significantly extend the known existence spectrum for $\text{PENT}\left(5,r\right)$.

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具有块尺寸3、4和5的五边形几何

五边形几何 $\text{便士}（ķ，\mathrm{[R}）$ 是部分线性空间，每行或每块都与 $ķ$ 点，每个点都与 $\mathrm{[R}$ 线，并针对每个点 $X$，有一条线入射恰好与这些线不在同一直线上的那些点 $X$。五边形几何形状中的相对线对由两条平行线组成，因此一条线中的每个点与另一条线上的那些点不共线。我们给出了具有块大小3的无限五边形几何图形和相连的缺陷图的直接构造。我们还提出了39个新的五边形几何图形，块大小为4，五个为五块大小为5，均具有连通的缺陷图。因此，我们确定了存在频谱，直至出现以下几种可能的例外情况为止：$\text{便士}（4，\mathrm{[R}）$ 不包含相反的线对，并且 $\text{便士}（4，\mathrm{[R}）$一对相反的线对。更普遍地，给定$Ĵ$ 我们表明存在一个 $\text{便士}（4，\mathrm{[R}）$ 和 $Ĵ$ 所有足够大的允许的相反的线对 $\mathrm{[R}$。使用块大小为5（包括类型）的一些新的组可分割设计${\mathrm{2个}}^{35}，{\mathrm{2个}}^{71}$， 和 $\mathrm{1个}{0}^{23}$）我们大大扩展了已知的存在范围 $\text{便士}（5，\mathrm{[R}）$。