Back in 2006 as a consequence of presenting my paper “Love, Understanding, and Soap-bubble foam”, and as plenary speaker at the “Bridges” Art/Science conference hosted by the University of London, I was invited to create an Art/Mathematics “Master-class” by the Royal Institution of Great Britain.
The Royal Institution’s motivation in asking me to create this module was to engage students with the wonder of Mathematics in an accessible and fun way.
My approach to this marvellous opportunity was to further develop my molecular modelling activities (using flexible straws/bonds in association with atom centres, as in my modelling of soap bubble foam).
My previous soap bubble modelling with the tetrahedral Carbon node
The molecular modelling node used in the foam project was the tetrahedral atom heart, as found in Carbon molecules. The geometry of this particular node is incredibly interesting, with its powerful structural attributes, but actually rather limited in its ability to model a wide variety of structures.
It didn’t take long to realise that the Cubi-lattice atom heart, as found in salt crystals, was full of exciting potential.
The cubic XYZ node
Now, the central issue with what became the “Bending X Y Z” system is very, very simple:
“Any polygon or polyhedron modelled using this node, other than a square, regular rectangle, cube or regular cuboid, will induce curvature”...
By shortening 2 edges curvature is induced
Many mathematical functions can be explored using this construction system: The Cartesian co-ordinate system, symmetries, projections, the Platonic solids with their relationship between angles, edges, and corners etc, etc.... But, my favourite area of investigation is that of percentages and the exponential.
Here is a module expressing percentages and the exponential in action:
Create an “Exponential” Tower using XYZ nodes and straws.
Create a ring of same length straws (I cut my first set of straws at 100mm so as to get at least two spars from each straw). Call this ring “A”.
I chose a pentagonal format for my ring, but you can choose another simple polygon or pattern if so desired.
Re-create the same ring of straws… but this time shorten the spar lengths by a percentage (subtracting about 5% works well). Call this ring “B”.
You now have two straw rings; ring “B” is 5% shorter in spar length than the first ring “A”.
Continue creating new rings which diminish in this way. So, for ring “C” subtract 5% from the spar length of ring “B”…and so on (until eventually the spar lengths are too short for the nodes to fit into).
To construct the Tower it is now time to join the rings in layers with “Spacer rods”.
These Spacer rods will grow in length by a percentage between each diminishing ring level (5% in this example is fine).
So…if we refer to the first Spacer rod lengths as length “A” it is possible to cut the length of the next set of Spacers (referred to as “B”) if we add 5% to their length.
Add 5% to the length of Spacer “B” to create the length of Spacer “C” and so on…do this between every level of straw rings.
VOILA...! You have created an Exponential Tower.
Now, take a look into a new world...
There can’t be too many of us who don’t feel the wonder of geodesic domes as have become famous through R. Buckminster-Fuller.
These truly amazing structures exude the qualities of optimisation and aesthetic grace. There are some great examples at the Eden Project, in the UK's lovely Cornwall.
The very first geodesic dome, designed by Walter Bauersfeld at Jena Germany, started prior to the first-world-war, interrupted by events, and finished in 1922.
But, even as supremely beautiful architectural structures they are limited by the rigour of the Platonic mathematics which generate them. Based upon the sub-division of an Icosahedron, they can express themselves with either triangular or a combination of pentagonal and hexagonal spar arrangements.
How could a geodesic dome be created outside of these constraints?
“BENDING XYZ” has an answer...
“It is possible to transform any 2-D linear network into a spherical shell, by employing a percentage differential between the original network and a clone of it, both being held apart at a consistent distance”.
The module below starts off with the example of a square lattice...purely to illustrate the idea as clearly as possible; but the possibilities are endless.
..too good to be true?
How to create a spherical network using the XYZ nodes and flexible straws…
By constructing 2 network designs (of the same pattern) and introducing “Spacer rods” to hold them a regular distance apart, it is possible to create spherical patterns.
The spherical design is made possible by the employment of “percentages”.
1/…Create a network/pattern using the straws and XYZ nodes. I kept it simple by making a grid. The straws of the initial “level” are cut to 100mm in length.
2/…Now, make a clone of your first pattern, but this time the length of the straws should be a percentage shorter than the original straw lengths…I suggest 10% shorter (90mm).
3/…Now you have created both patterns you can link them together by the use of “Spacer rods”. These are straws cut to a regular length to hold the two levels above and below each other. For my grid I cut the spacer rods to a length of 25mm, but this is optional.
Once this activity is well understood, irregular lengths within a pattern can be employed for more adventurous results.
Below, I have created a whole sphere, in this case I used 120 0 nodes, in line with the 2-D foam structure I was interested in.