#
On Maximal Homogeneous 3-Geometries and Their Visualization^{ †}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

- (i)
- These (commutative group of) transforms commute with the matrix of Jordan form$$\begin{array}{c}\left(\begin{array}{cccc}1& 1& 0& 0\\ 0& 1& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\end{array}$$$${(ds)}^{2}={(dx)}^{2}+{[(dx)(-x)+(dy)]}^{2}+{(dz)}^{2}={(dx)}^{2}(1+{x}^{2})2(dx)(dy)x+{(dy)}^{2}+{(dz)}^{2};\phantom{\rule{0ex}{0ex}}\begin{array}{c}\mathrm{leads}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{3.33333pt}{0ex}}{x}^{\u2033}=0,\phantom{\rule{3.33333pt}{0ex}}{y}^{\u2033}-{x}^{\prime}{x}^{\prime}=0,\phantom{\rule{3.33333pt}{0ex}}{z}^{\u2033}=0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{so}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{3.33333pt}{0ex}}{x}^{\prime}(0)=u,\phantom{\rule{3.33333pt}{0ex}}{y}^{\prime}(0)=v,\phantom{\rule{3.33333pt}{0ex}}{z}^{\prime}(0)=w,\\ {u}^{2}+{v}^{2}+{w}^{2}=1,\phantom{\rule{3.33333pt}{0ex}}\mathrm{to}\phantom{\rule{3.33333pt}{0ex}}x=us,\phantom{\rule{3.33333pt}{0ex}}y=\frac{1}{2}{u}^{2}{s}^{2}+vs,\phantom{\rule{3.33333pt}{0ex}}z=ws;\end{array}$$
- (ii)
- These transforms commute with the matrix$$\begin{array}{c}\left(\begin{array}{cccc}1& 1& 0& 0\\ 0& 1& 1& 0\\ 0& 0& 1& 1\\ 0& 0& 0& 1\end{array}\right)\end{array}$$$$\begin{array}{c}{(ds)}^{2}={(dx)}^{2}+{[(dx)(-x)+(dy)]}^{2}+{[(dx)({x}^{2}-y)+(dy)(-x)+(dz)]}^{2}\\ ={(dx)}^{2}(1+{x}^{2}+{x}^{4}-2{x}^{2}y+{y}^{2})-2(dx)(dy)(x+{x}^{3}-xy)+2(dx)(dz)({x}^{2}-y)\\ +{(dy)}^{2}{(1+x)}^{2}-2(dy)(dz)x+{(dz)}^{2};\phantom{\rule{3.33333pt}{0ex}}\mathrm{leads}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\end{array}$$$$\begin{array}{c}{x}^{\u2033}=0,\phantom{\rule{3.33333pt}{0ex}}{y}^{\u2033}-x{x}^{\prime}{x}^{\prime}=0,\phantom{\rule{3.33333pt}{0ex}}{z}^{\u2033}+x{x}^{\prime}{x}^{\prime}-2{x}^{\prime}{y}^{\prime}=0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{so}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{3.33333pt}{0ex}}{x}^{\prime}(0)=u,\phantom{\rule{3.33333pt}{0ex}}{y}^{\prime}(0)=v,\phantom{\rule{3.33333pt}{0ex}}{z}^{\prime}(0)=w,\phantom{\rule{3.33333pt}{0ex}}\\ {u}^{2}+{v}^{2}+{w}^{2}=1,\phantom{\rule{3.33333pt}{0ex}}\mathrm{to}\phantom{\rule{3.33333pt}{0ex}}x=us,\phantom{\rule{3.33333pt}{0ex}}y=\frac{1}{2}{u}^{2}{s}^{2}+vs,\phantom{\rule{3.33333pt}{0ex}}z={u}^{3}/6{s}^{3}+uv{s}^{2}+ws;\end{array}$$

## 2. Specific Geometries, ${\mathbf{H}}^{2}\times \mathbf{R}$ and $\tilde{\mathbf{S}{\mathbf{L}}_{2}\mathbf{R}}$ in Models

## 3. Crystallography in Non-Euclidean Spaces

#### 3.1. Hyperbolic Space ${\mathit{H}}^{\mathbf{3}}$

**Theorem**

**1.**

#### 3.2. $\mathit{Nil}$ Space

#### 3.3. $\mathit{Sol}$ Space

#### 3.4. ${\mathit{S}}^{\mathbf{2}}\times \mathit{R}$ Space

## 4. Conclusions

## Conflicts of Interest

## References

- Molnár, E. The projective interpretation of the eight 3-dimensional homogeneous geometries. Beitr. Algebra Geom. Contrib. Algebra Geom.
**1997**, 38, 261–288. [Google Scholar] - Molnár, E.; Szirmai, J. Symmetries in the 8 homogeneous 3-geometries. Symmetry Cult. Sci.
**2010**, 21, 87–117. [Google Scholar] - Molnár, E.; Prok, I.; Szirmai, J. The Euclidean visualization and projective modelling the 8 Thurston geometries. Stud. Univ. Zilina Math. Ser.
**2015**, 27, 35–62. [Google Scholar] - Molnár, E.; Prok, I.; Szirmai, J. Classification of tile-transitive 3-simplex tilings and their realizations in homogeneous spaces. In Non-Euclidean Geometries, János Bolyai Memorial Volume, Mathematics and Its Applications; Prékopa, A., Molnár, E., Eds.; Springer: Boston, MA, USA, 2006; Volume 581, pp. 321–363. [Google Scholar]
- Molnár, E.; Szirmai, J.; Vesnin, A. Packings by translation balls in $\tilde{\mathbf{S}{\mathbf{L}}_{2}\mathbf{R}}$. J. Geom.
**2014**, 105, 287–306. [Google Scholar] [CrossRef] - Molnár, E.; Szirmai, J.; Vesnin, A. Geodesic and translation ball packings generated by prismatic tessellations of the universal cover of
**SL**_{2}**R**. Results Math.**2017**, 71, 623–642. [Google Scholar] [CrossRef] - Molnár, E.; Szirmai, J. Classification of
**Sol**lattices. Geom. Dedicata**2012**, 161, 251–275. [Google Scholar] [CrossRef] - Molnár, E.; Szirmai, J. Top dense hyperbolic ball packings and coverings for complete Coxeter orthoscheme groups. arXiv, 2017; arXiv:161204541v1. [Google Scholar]
- Prok, I. Classification of dodecahedral space forms. Beitr. Algebra Geom. Contrib. Algebra Geom.
**1998**, 39, 497–515. [Google Scholar] - Molnár, E. Polyhedron complexes with simply transitive group actions and their realizations. Acta Math. Hung.
**1991**, 59, 175–216. [Google Scholar] - Molnár, E.; Szirmai, J. On hyperbolic cobweb manifolds. Stud. Univ. Zilina Math. Ser.
**2016**, 28, 43–52. [Google Scholar] - Szirmai, J. The optimal ball and horoball packings to the Coxeter honeycombs in the hyperbolic d-space. Beitr. Algebra Geom. Contrib. Algebra Geom.
**2007**, 48, 35–47. [Google Scholar] - Kellerhals, R. On the volume of hyperbolic polyhedra. Math. Ann.
**1989**, 245, 541–569. [Google Scholar] [CrossRef] - Szirmai, J. The densest geodesic ball packing by a type of
**Nil**lattices. Beitr. Algebra Geom.**2007**, 48, 383–397. [Google Scholar] - Molnár, E.; Szilágyi, B. Translation curves and their spheres in homogeneous geometries. Publ. Math. Debr.
**2011**, 78, 327–346. [Google Scholar] [CrossRef] - Scott, P. The geometries of 3-manifolds. Bull. Lond. Math. Soc.
**1983**, 15, 401–487. [Google Scholar] [CrossRef] - Szirmai, J. The densest translation ball packing by fundamental lattices in
**Sol**space. Beitr. Algebra Geom. Contrib. Algebra Geom.**2010**, 51, 353–373. [Google Scholar] - Cavichioli, A.; Molnár, E.; Spaggiari, F.; Szirmai, J. Some tetrahedron manifolds with
**Sol**geometry. J. Geom.**2014**, 105, 601–614. [Google Scholar] [CrossRef] - Szirmai, J. A candidate to the densest packing with equal balls in the Thurston geometries. Beitr. Algebra Geom.
**2014**, 55, 441–452. [Google Scholar] [CrossRef] - Molnár, E. Two hyperbolic football manifolds. In Proceedings of the International Conference on Differential Geometry and Its Applications, Dubrovnik, Yugoslavia, 26 June 1988; pp. 217–241. [Google Scholar]
- Molnár, E. Projective Metrics and hyperbolic volume. Ann. Univ. Sci. Budap., Sect. Math.
**1989**, 32, 127–157. [Google Scholar] - Molnár, E. Combinatorial construction of tilings by barycentric simplex orbits (D symbols) and their realizations in Euclidean and other homogeneous spaces. Acta Cryst.
**2005**, A61, 542–552. [Google Scholar] [CrossRef] [PubMed] - Prok, I. Data structures and procedures for a polyhedron algorithm. Period. Polytech. Ser. Mech. Eng.
**1992**, 36, 299–316. [Google Scholar] - Szirmai, J. Geodesic ball packing in
**S**^{2}×**R**space for generalized Coxeter space groups. Beitr. Algebra Geom. Contrib. Algebra Geom.**2011**, 52, 413–430. [Google Scholar] [CrossRef] - Szirmai, J. Lattice-like translation ball packings in
**Nil**space. Publ. Math. Debr.**2012**, 80, 427–440. [Google Scholar] [CrossRef] - Weeks, J.R. Real-time animation in hyperbolic, spherical, and product geometries. In Non-Euclidean Geometries, János Bolyai Memorial Volume, Mathematics and Its Applications; Prékopa, A., Molnár, E., Eds.; Springer: Boston, MA, USA, 2006; Volume 581, pp. 287–305. [Google Scholar]
- Wildberger, N.J. Universal hyperbolic geometry, sydpoints and finite fields: A projective and algebraic alternative. Universe
**2017**. submitted. [Google Scholar]

**Figure 1.**Our scene for dimensions 2 with projective sphere $\mathcal{P}{\mathcal{S}}^{2}$ embedded into the real vector space ${\mathbf{V}}^{3}$ and its dual ${\mathit{V}}_{3}$.

**Figure 2.**The hyperbolic plane ${\mathbf{H}}^{2}$, embedded into ${\mathcal{P}}^{2}\subset \mathcal{P}{\mathcal{S}}^{2}$ by a conic polarity $u(\mathit{u})\to U(\mathbf{u})$, $p\to P$, $a\to A$ (the Beltrami–Cayley–Klein disc model). Here, we illustrate the projective model of ${\mathbf{H}}^{2}\times \mathbf{R}$ geometry, too. Imagine similarities from the origin O. The logarithm of similarity factor will be the $\mathbf{R}$-parameter.

**Figure 3.**The unparted hyperboloid model of $\tilde{\mathbf{S}{\mathbf{L}}_{2}\mathbf{R}}=\tilde{\mathcal{H}}$ of skew line fibres growing in points of a hyperbolic base plane ${\mathbf{H}}^{2}$. In addition, a Gum-fibre model of Hans Havlicek and Rolf Riesinger, used also by Hellmuth Stachel with other respects (Vienna UT).

**Figure 7.**The densest lattice-like congruent ball configuration in $\mathbf{Nil}$ space related to the lattice parameter $k=1$, the central red ball is touching 14 surrounding green ones (computer picture by Benedek Schultz), and a horizontal geodesic curve lying in a hyperbolic paraboloid.

**Figure 8.**Fundamental lattice in $\mathbf{Sol}$ geometry. Notice the Minkowskian base lattice of special relativity.

**Figure 10.**The conjectured densest geodesic ball packing configuration for all Thurston geometries in the specific projective model of ${\mathbf{S}}^{2}\times \mathbf{R}$ geometry whose density is ≈0.87757⋯

**Table 1.**Thurston geometries each modelled on $\mathcal{P}{\mathcal{S}}^{3}$ by specified polarity or scalar product and isometry group.

Space X | Signature of Polarity $\mathit{\Pi}({}_{\u2605})$ or Scalar Product $\langle \phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\rangle $ in ${\mathit{V}}_{4}$ | Domain of Proper Points of X in $\mathcal{P}{\mathcal{S}}^{3}$ $({\mathbf{V}}^{4}(\mathbf{R}),\phantom{\rule{0.166667em}{0ex}}{\mathit{V}}_{4})$ | The Group $\mathit{G}=\mathbf{Isom}\mathbf{X}$ as a Special Collineation Group of $\mathcal{P}{\mathcal{S}}^{3}$ |
---|---|---|---|

${\mathbf{S}}^{3}$ | $(+\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+)$ | $\mathcal{P}{\mathcal{S}}^{3}$ | Coll $\mathcal{P}{\mathcal{S}}^{3}$ preserving $\Pi ({}_{\u2605})$ |

${\mathbf{H}}^{3}$ | $(-\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+)$ | $\{(\mathbf{x})\in {\mathcal{P}}^{3}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}\langle \mathbf{x},\mathbf{x}\rangle <0\}$ | Coll ${\mathcal{P}}^{3}$ preserving $\Pi {(}_{\u2605})$ |

$\tilde{\mathbf{S}{\mathbf{L}}_{2}\mathbf{R}}$ | $(-\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+)$ with skew line fibering | Universal covering of $\mathcal{H}:=\{\left[\mathbf{x}\right]\in \mathcal{P}{\mathcal{S}}^{3}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}\langle \mathbf{x},\mathbf{x}\rangle <0\}$ by fibering transformations | Coll ${\mathcal{P}}^{3}$ preserving $\Pi {(}_{\u2605})$ generated by plane reflections |

${\mathbf{E}}^{3}$ | $(0\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+)$ | ${\mathcal{A}}^{3}={\mathcal{P}}^{3}\backslash \left\{{\omega}^{\infty}\right\}$ where ${\omega}^{\infty}:=({\mathit{b}}^{0})$, ${\mathit{b}}_{\u2605}^{0}=\mathbf{0}$ | Coll ${\mathcal{P}}^{3}$ preserving $\Pi {(}_{\u2605})$, generated by plane reflections |

${\mathbf{S}}^{2}\times \mathbf{R}$ | $(0\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+)$ with O-line bundle fibering | ${\mathcal{A}}^{3}\backslash \left\{O\right\}$, O is a fixed origin | G is generated by plane reflections and sphere inversions, leaving invariant the O-concentric 2-spheres of $\Pi {(}_{\u2605})$ |

${\mathbf{H}}^{2}\times \mathbf{R}$ | $(0\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+)$ with O-line bundle fibering | ${\mathcal{C}}^{+}=\{X\in {\mathcal{A}}^{3}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}$$\langle \overrightarrow{OX},\overrightarrow{OX}\rangle <0$, half cone} by fibering | G is generated by plane reflections and hyperboloid inversions, leaving invariant the O-concentric half-hyperboloids in the half-cone ${\mathcal{C}}^{+}$ by $\Pi {(}_{\u2605})$ |

$\mathbf{Sol}$ | $(0\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}+)$ and parallel plane fibering with an ideal plane $\varphi $ | ${\mathcal{A}}^{3}={\mathcal{P}}^{3}\backslash \varphi $ | Coll. of ${\mathcal{A}}^{3}$ preserving $\Pi {(}_{\u2605})$ and the fibering with 3 parameters |

$\mathbf{Nil}$ | Null-polarity $\Pi {(}_{\u2605})$ with parallel line bundle fibering F with its polar ideal plane $\varphi $ | ${\mathcal{A}}^{3}={\mathcal{P}}^{3}\backslash \varphi $ | Coll. of ${\mathcal{A}}^{3}$ preserving $\Pi {(}_{\u2605})$ with 4 parameters |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Molnár, E.; Prok, I.; Szirmai, J. On Maximal Homogeneous 3-Geometries and Their Visualization. *Universe* **2017**, *3*, 83.
https://doi.org/10.3390/universe3040083

**AMA Style**

Molnár E, Prok I, Szirmai J. On Maximal Homogeneous 3-Geometries and Their Visualization. *Universe*. 2017; 3(4):83.
https://doi.org/10.3390/universe3040083

**Chicago/Turabian Style**

Molnár, Emil, István Prok, and Jeno Szirmai. 2017. "On Maximal Homogeneous 3-Geometries and Their Visualization" *Universe* 3, no. 4: 83.
https://doi.org/10.3390/universe3040083