This is like a
Zome, except that the struts are forced outwards more, so that the resulting
shape is spherical.
The benefits of this dome over a normal geodesic dome are:
1) All non horizontal struts are the same length. In the dome here in Richmond Park there are no horizontal struts, so all struts are the same length.
2) Unlike geodesic domes, where there are at most a couple of horizontal levels suitable to meet the ground, this dome is naturally horizontally layered at every level.
3) It folds up into a neat bundle. To fold this dome up I just unhooked 6 threads.
This dome is six sided around the apex. I call the apex the "north pole". The next level of nodes down I call the "arctic", then the next level down is the "tropic", and finally the nodes at ground level are the "equator". Strangely, the "squares" you can see are not really square, but are ever so slightly deformed.
The arctic, tropical, and equatorial
circles are
tensioned by lengths of invisible nylon. There are also six strands connecting the
north pole to the tropical level. Without these tensioning
elements the whole thing would fall down. Originally I tried hanging a pineapple
from the apex to achieve the same effect as these extra six tensioning
elements. In theory they would prevent the dome from collapsing inwards.
But that is another story.
Though the bamboo struts are all the same length, the nylon strands are at different length at each level. For some mysterious (to me at least) reason root(13) seems to be the key unit of measurement.
In calculating the dimensions, I assume the radius of the dome is 1, and then calculate the length of the poles and strands. Then to actually build it I have to divide every thing by the theoretical pole length, and then multiply everything by the actual pole length.
So here are the calculations for the exact length of the dome parts when the radius is 1. If you can think of a simpler method I would like to hear about it.
To
calculate the dimensions imagine the total spherical framework.6 struts radiate from each pole.
The ends of these form the "arctic" circles.
To the ends of these are connected 12 struts,
the lower ends of which define the "tropic".
Then a further layer of 12 struts extends the
structure to the equator.
Now take a profile snapshot so that the strut linking the pole P
to the arctic circle at A is in the plane of
the snapshot.
The diagram shows a part of this snapshot with the pole at P,
a strut leading to the arctic at A, another
strut leading to the tropic at T, and another
strut down to the equator at E.
X is a node point on the southern tropic south of T.
The radius is 1 and the center is at O.
The length i is the vertical distance from
P to the arctic circle.
Notice, by symmetry that i is also the
distance from A to the line
PT, and also the distance from E to
the line TX.
Note that AT=TE. Then since PA=EX,
again by symmetry PT = TX.
Note also TX is parallel to PO.
Angle <EOX = angle <POA. Since
PT is at right angles to OA
we can deduce that PT is parrallel to TX.
Recap: PT=TX, PO//TX OX//PT.
So OPTX is a parallelogram
of side 1. Therefore PT=1.
This next diagram has vertical and horizontal distances defined
for P and T. R=a+b
is the radius at the tropic. Using the fact that
PT = 1, and pythagoras' theorem, we can get
the following simultaneous equation:
Which
solves to 
The
diagram on the left here can be used to calculate the length e
in terms of a, which incidentally is 1-i.
The diagram on the right shows a view from above and explains the relation between a and R.
Next
diagram is the same as the original snapshot but with more struts shown.
The dotted lines are tension strands. Using pythagoras' theorem and the
property of the hexagon/12-gon as shown above,the lengths can be calculated as below.
Note that g=1 because it is one side of a hexagon of radius 1 at the equator. For a similar reason K=R.
The
length properties on the left here can be evaluated and solved to get the
values on the right.
Now we have all the relative lengths of all segments it is "easy" to build the dome and relax inside.
