Leonardo Sticks History
Leonardo da Vinci (1452-1519) invented a great many machines to do an extraordinary number of things, and in many cases it has taken centuries for the rest of us to catch up with him. He sketched ideas for helicopters, submarines, parachutes, and bicycles long before anyone else thougt they were practical.
On two pages of sketches Leonardo described a roofing system for spanning large areas without internal support. He shows wooden beams laced together in a particular way so that they are self-supporting, and says this idea can be used to cover a space without internal support, quickly and simply without complicated joints or special tools. The structures are shallow domes that are built starting from a center, supporting themselves on the ends of new sticks added to the edges. Leonardo says that the beams should be tied together with ropes and covered with strips of woven wool. He probably had in mind a shady cover for a space like a marketplace or military camp. There is no record that any of them were ever built.
In 1989 Dutch sculptor Rinus Roelofs was working on ways to divide a sphere and found a system that was simple and elegant. He recognized that in addition to dividing a sphere into solid pieces, he could also make the joints of that division into wooden sticks that interlaced to form the sphere. He invented sticks with two notches to help with alignment. They didn't need to be tied together as Leonardo's beams did: their weight alone held them in place. Unable to believe that he was the first to find such a geometry, he went to his local library and eventually found the page of Leonardo's notebooks with the sketches, which he recognized as the same as his own system. He contacted Kim Williams, editor of the Nexus Network Journal for Architecture and Mathematics, and she invited him to share his discovery with a number of architects and mathematicians in Italy in June 2003. The group assembled several of Leonardo's structures and studied the relationship to other similar structures.
Maine architect Christopher Glass teamed up with Rinus, and Leonardo Sticks were born. The sticks are Rinus' small models of the large beams Leonardo described.
The closest parallels to Leonardo's ideas are those of Buckminster Fuller. Fuller spent his life developing systems that made lightweight structures from repeated elements, and at the end of his life was working on what he called "deresonated tensegrity spheres" which were one version of Leonardo's idea. Leonardo's system, however, has many variations that went beyond the geometry of Fuller's domes.
The One Rule
Rinus Roelofs developed a simple way of describing the one Rule that governs the building of all the structures. He called it "+ - - +", or the "plus - minus - minus - plus" Rule. This means that each stick in a structure is connected at four points to four other sticks, and the way they connect depends on which side - top or bottom - the other sticks are on. A stick's end must be over the notch of another stick, and each notch in the middle must be under the end of another stick. So another way of reading the Rule is "over - under - under - over". A useful reminder is that the notches in a stick must always face up, to receive the ends of other sticks on top.
Rinus used this Rule to build not only the three or four patterns that Leonardo steched, but over one hundred additional patterns. In all of them each stick observes the + --+ rule, but the results can be very different A few of these patterns are included in the kit, but there are lots of others.
Starting a Dome
It is best to build your model on a carpeted floor or a table with a tablecloth. Wood floors and tables can be too slippery.
All patterns have to have a starting place, and we recommend two basic starting shapes, the triangle and the square. Other shapes can be used to start, but these make it easiest to keep the pattern neat and correctly aligned. The triangle is the simpler: place the end (+) of a second
stick on top of one of the notches (-) on the first stick Now take a third stick, place its end (+) on the second notch on the second stick lift the end of the first stick and swing the third stick under it until its notch is under it. Lower it into the notch. The triangle so formed will now be supported in the air by the outer ends of the three sticks. This is a "two-unit" triangle, a unit being the distance between the two notches. The end of a stick should overlap the one below by about half a unit, so that the spacing stays even.
The square works the same way, except that you put three sticks at right angles before adding the fourth to complete the square. Shapes built from squares are usually more stable than ones built from triangles.
That's all there is to building all the structures, and each new stick is added in the same way. Place the end of a new stick on a notch of another stick, and lift whichever stick will rest on a notch of the new stick The trick is not getting lost in the pattern, and especially in sticking to the Rule: always over-under-under-over.
Leonardo's Patterns
Leonardo's Codex Atlanticus has only two pages (Folio 899 verso) on this idea. The pages are pretty messy and hard to read. Our close study has identified three definite patterns and hints for a fourth, along with Leonardo's notes and sketches for two other ideas that he didn't develop.
You can find these patterns as well as many others at the 'examples' page.