## Aperiodic Rhombohedra

In the Form of a Chiral Lattice – Initial Public Rendering, April 3, 2023

### Analysis Animation

Explaining Layers, Rhombohedral Geometry & Codes

No Cracks, No Overlaps

### Description

This is an aperiodic rhombohedral lattice without gaps or overlaps. It is made of a unique lattice element, itself made of five unique layers with five rhombohedra in each layer*. Having 36-fold symmetry each lattice element is repeated 35 times, rotated 10° about the ‘center’. Each rhombohedra in each layer is a member of the rhombi combination series of that layer. Each series/layer has one constant rhomb member across the series. All rhombohedra contain three pairs of identical rhombi and all are chiral. Alternate layer series are distinct in form. I call them ‘Bricks’ and ‘Blades’. Gaps between “Bricks” are filled with ‘Blades’ layers. The next layer of ‘Bricks’ is then ‘laid’ upon the previous ‘Bricks’ layer leaving a gap to be filled with the next ‘Blade’ layer, etc., etc. ‘Blades’ layers help to tie ‘Brick’ layers together.

### Bricks & Blades Aesthetic Animation

Aperiodic Rhombohedral Chiral Lattice as Beauty

### A Note on the Vertices

Significance of Vertical Stratification

Orthographic Side View Showing All Vertices and Edges (from fading Surface View)

This class of rhombohedra all possess a crucial property in addition to all edges being the same length. That is, every edge of all rhombs in this class all possess the same angle relative to the axis of rotation, or the plan / top view. The consequence of this is that all vertices of this aperiodic lattice exist in parallel planes with identical vertical distances between each plane and its neighbors.

### Largest Element & Rhombohedral Details

Rhombohedra Specifics

Names, Codes and Top / Plan View Top Vertex Angles of the 25 Rhombohedra which make up the Aperiodic Lattice

This image contains properties of the 25 rhombohedra in the lattice I developed.(T1R1 through T5R5). In the image above some of the cells are given the same colors. These are geometrically identical. Interestingly, each identical pair is made of one ‘Brick’ and one ‘Blade’ rhombohedra. There is an obvious pattern vaguely reminiscent of the periodic table of the elements. The list is not complete, but I have been unable to discover why each entry appears to be duplicated in another tier type (Brick or Blade), or why no 3rd or 4th, etc. identical rhombohedra seem to appear. More ‘twins’ become apparent in the expanded view which includes additional potential tiers and entries in each tier that I did not develop.

#### In the table below, T = Tier and R = Rhombohedra.

- ‘Twin’ TR Values
- T1R1, T2R1
- T2R2, T3R1
- T3R2, T4R2
- T4R3, T5R2
- T1R2, T4R1
- Twins 1 In, 1 Out
- T5R3, T6R3
- T1R3, T6R1
- T3R3, T6R2
- T1R4, T8R1
- T2R4, T7R1

- Rhombohedra Code
- 011817
- 011716
- 021715
- 021614
- 021819
- Rhombhedra Code
- 031613
- 031815
- 031714
- 041814
- 011514

- Rhombs Apex ∠s*
- 10° 180° 170°
- 10° 170° 160°
- 20° 170° 150°
- 20° 160° 140°
- 20° 180° 160°
- Rhomb Apex ∠s*
- 30* 160* 130*
- 30* 180* 150*
- 30* 170* 140*
- 40* 180* 140*
- 10* 150* 140*

### Prior Work

Leading to Aperiodic Chiral Rhombohedra Lattice

#### Muqarnas Block Dome

This is my first creation of the Bricks and Blades aperiodic space filler. It dates to March 1, 2015. Unfortunately I had no idea that it was an aperiodic 3D space filler. As can be seen I created both a single Brick layer and a single Blade layer.. I didn’t realize that they could be followed by more Bricks and Blades. So I also had no idea that they would fill space without cracks or overlaps. Although they use the same four rhombic vertices, I used those vertices to define a pair of three point NURBS surfaces, instead of flat rhombic faces. This made their combinations visually more interesting. An animation can be viewed here.

Additionally, this form uses 32-fold symmetry. So the plan view apex angles are in increments of 360°/32 or 11.25° instead of the 10° plan view increment of 36-fold symmetry. Furthermore, this is a 16-fold symmetry form and uses the ‘Ray‘ motif expansion method. I believe that the bilaterally symmetrical overall form and characteristics are a consequence of ‘halving’ the symmetry from 32-fold to 16-fold. The form on the top of this page uses the ‘Crescent‘ motif expansion method. This is demonstration of the very wide variety of aperiodic 3D lattices with the higher order of N-fold symmetries. For a look at 36-fold motif variation see 36-fold and 32-fold.

The aperiodic lattice above is one of a pair of chiral forms (right- and left-handed), whereas the earlier form displays rotational and mirror symmetry. The form above is made entirely of chiral rhombohedra while the earlier form is made of both bilaterally symmetrical and chiral rhombohedra.

#### Papyrus Form

This is my first foray into using hexahedral mogharnas blocks in a composition. It dates to November 23, 2012. An animation can be viewed here.

In an effort to further explore possible symmetrical form I pushed mogharnal faces in towards the center or out away from it. I found that, at the right position, it was possible to always add four muqarnas from its symmetry group that would seamlessly join the translated tile to its original position. This was truly my first awareness of and use of hexahedral mogharnas blocks. I later used planar figures to connect the four vertices of the muqarnas surface sets. At that point it became evident that all faces were rhombs and that the hexahedral blocks were in fact rhombohedra.

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