un espace pour les homme       



Geometry (continued)

From Zonaedron to Zome

The zonaèdre is a mathematical, unreal being
The zome is a constructed zonaedreon
From one to the other, choices are made (materials and construction techniques, habitability, etc.)


Axis position
Generally, the axis of the zome is positioned vertical, but it is not an obligation
Vertical, it leads to aesthetic, well-balanced volumes and great rigidity
But we can also position this oblique axis, or even horizontal

   
             

Zonaedron 6 with vertical axis, inclined (2 positions) and horizontal (2 positions)

Cups
For reasons of stability and habitability, the zome is only a portion of zonaedron
We must therefore choose a section plane, either the equator plane for even order zomes, or the plane below the equator for odd numbers.
In these 2 cases, we may or may not extend the zome by vertical walls
In even zomes, the crown at the equator is vertical (parallel to the axis), this is not the case for odd numbers

                          
Zome 6 cut under the equator and resting on triangles Zome 6 cut at the equator and extended by walls
N * N/2 diamonds = 18
N triangles = 6
N * (N/2 - 1) ldiamonds = 12
N triangles+walls = 6



Zome 5 cut under the equator and resting on triangles Zome 5 cut above the equator and extended by walls
N * (N - 1) /2 ldiamonds = 10
N triangles = 5
N * (N - 1) /2 diamonds = 10
2 * N walls = 10
             
Dimensions
For a given edge length, the higher the order number, the larger the zome
The following table gives for various N the (approximate) dimensions obtained with an edge length of 2 m and two of the most common shape numbers. These are maximum dimensions not to be exceeded. Reduce the edge to stay below.

For odd zomes 5 and 7, these are the dimensions on the ground for a zome without walls (re-entering)

N Side Diameter Surface
4 3,40 - 3,58 4,80 - 5,05 11,60 - 12,80
5 3,20 - 3,40 5,00 - 5,20 18,00 - 19,90
6 3,40 - 3,58 6,80 - 7,15 30,00 - 33,25
7 3,30 - 3,50 7,25 - 8,00 40,00 - 44,20
8 3,40 - 3,58 8,90 - 9,35 55,90 - 61,80
10 3,40 - 3,58 11,00 - 11,55 89,00 - 98,5
12 3,40 - 3,58 13,15 - 13,80 129,50 - 143

Extensions

These are volumes that are added to enlarge the living space (or create separate rooms)
This is easy with even-order zomes with vertical walls
   

        Extension with roof with 1 slope and roof with 2 slopes


Extension by part of zome and by grouping with another zome
Zones
A continuous band of facets which surrounds the volume is called the zone. One of the edges of the facets remains parallel to a direction. These zomes are the seat of spiral zones. A facet is created by the intersection of two zones.


In "classic" zomes, there are as many zones as the zome order. However, one can add or suppress one or more zones of a zonaèdre, or else deform them: we will still obtain a zome.
Two types of frequently encountered zones: (for example on this zome 6)



Zone M: the plane is cut along a median, the two halves separated, and a zone is introduced between the two parts


Zone D: the plane is cut along a diagonal, the two halves separated, and one is zone introduced between the two parts

The introduction of one or other of these zones makes it possible to enlarge the surface of the zome without increasing either the complexity or the height of the volume. Hexagons are created at the intersection of 3 zones.
Previous Reception