IC16: The Self-Sustaining Domes By Leonardo (Extended Talk)
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Brasó, Enric
Roelofs, Rinus
Hidden in a corner of the writings of Leonardo da Vinci, there are sketched two structures of self sustaining domes made from sticks without any clamping element. Rediscovered and studied by Rinus Roelofs, recently a work group from the MMACA Association has developed on this concept and designed 11 dif- ferent patterns. We are taking this activity to schools, museum workshops and street fairs. This activity is spectacular, attractive and adaptable to all ages, we can even work with groups of five year old.
 Music In about an hour we are going to build a very big dome outside just near this room. But before that we will give two different presentations. One about real experiences with dome building done by Andriik. And I will start with a small introduction about the history of this complete project. For me it all started with this object. Well it does look like a dome I agree. It's a spherical concession built with just paper. But this was my first object that led me to the dome concessions. And this object has a lot to do with the section of a cube. So I started examining how you could make nice dissections of cubes in different parts. But at a certain moment I was more concentrated on the lines on the phases of the cubes than the parts itself. So when you take those lines and you extend them to phases you get an object. And when you round off all the corners you get this spherical concession that you have seen. And well I built models in wood. And well this was the first step for me to come to the dome concessions. Because when you make the mathematical representation of this object so you only draw the lines and the connections a graph then you see that the connecting lines have all the same quality. They are all connected. Each line is connected to four other lines. And in a special way because all the endpoints of each of the lines so the endpoints of this line are connected to one of the midpoints of other lines. So this is the basic definition of this kind of concession. And I call this positive, negative, negative, positive. So all the elements have this property and can be connected in this way. And I built my first sculpture in this way. This is on a school. But also here you can see each element is connected with four others. And endpoints always meet midpoints of other objects. I found this so interesting that I studied the possibilities of this system a little bit further. And I came to the conclusion that if you change positive and negative to over and under like I did here in this drawing, that in fact this is a description of a 3D concession. A small dome as you can see. And when you extend the drawing with more elements, well, you'll get a bigger dome. And finally this was my first dome. Well, you can see it's created in, well, about 25 years ago. So this is a very old picture. But for me this was the start to dome building. So starting from Svirs, I stepped a little bit further. And studying the system of positive, negative, negative, positive, it led to a lot of different patterns that I could use to build domes. And all kinds of domes could be created with the same elements. The same very simple wooden elements on which I made two incisions to have the positive, negative, negative, positive structure. A variety of different patterns, regular but also more irregular. Here with three pentacons in the middle and, well, the possibilities grow. And well, that was the moment that I thought, well, this system is so nice and so simple that I couldn't imagine I was the first one to use this to make this. It was 1990, so at that time reciprocal structures weren't very well known. Also internet wasn't available for me to search for other examples. So I had to go to libraries and I finally found this page on which I could see drawings that are, well, almost the same as I made. And this page is a page from a notebook of Leonardo da Vinci. So a very old source and, well, I had to admit, well, when you compare his drawings to mine, well, they're almost the same but he was the first. And also this one, well, this was my drawing, but yeah, I have to admit, he was the inventor of this system. Well, I explored the possibilities of these kinds of elements with the connections as I told you and I was able to make a lot of different patterns. You see six patterns here, different patterns with all the same properties, but in total I've got more than 100 different patterns. Well, I'm from Holland, like M.G. Escher, so I explored some more possibilities of these patterns and I think this comes close to Escher. So when I discovered the variety of possibilities with the simple sticks or the simple elements, the simple positive, negative, negative, positive elements, I went on searching for other things and one of the things is that you can distort a pattern without leaving the system because, again, every element is connected in the right way. So you can make nice distortions and, well, I can show you an animation of such a distortion. You start with a pattern like this and let's say you take a few elements in the middle and you just rotate them a little bit. Then you end up with a distortion but the system is followed in a perfect way and in real, the distortion in the pattern looks like this. Well, when you can make a distortion at one point, you can do it all over. So you can change from one pattern to another in this way. There are more transformations I discovered but I will leave it by this one. So from one pattern you can create a new one. Well, I talked about Escher, so for me it was very simple to take the step to tiles. So this line is an element of a dome but it's also a line in between the tiles in this case but the pattern is exactly the same. But when you take the step to tiles, you see other possibilities and one of them is this one. I can change the tiles a little bit and now I can transform the pattern from left turning to right turning. And there's much more that you can do now because when you look at the line that's in between those tiles, well, you can say this is a simple line but you can also say this line is the edge of this tile and also the edge of this tile and the edge of this tile and the edge of this tile. So in fact this line consists of four different lines, well, like this. And because of the system, these lines are always equal length and also these lines are also of equal length. So in fact you've got a parallelogram and that makes it possible to add some movement to the system like this. Just because you use the positive, negative, negative, positive element in your pattern. And well, this has a lot of possibilities. This is another pattern of the grid, the grid that can be used for a dome but also be transformed in tiles and then it still works just because of the simple element. And you can even go further. When I've got a point with a hexagon here, I can move the bars a little bit away from the center. So I got space for new smaller bars to make a smaller hexagon, so a fractal step. And when you translate it into tiles, well, again, the transformation still works. So this gives a lot of dynamic systems with simple tiles all based on the patterns with the plus, minus, minus, plus structure. So here's the last one. So this is, you can see this line is a line that you can use for a dome, but with the tiles I can make this transformation. And you can see it transforms into another grid that also has the same quality for the domes. So with the four points. So this was one field of exploration for me. And another field based on this pattern was, well, when you superimpose another grid on the first grid, you can use the blue grid now to make a polyhedron. Those triangles can be seen as phases of a polyhedron, but now this grid is the decoration of the final polyhedron. So when I fold it together, let's say to an octahedron, I have the grid completely around this octahedron. So a three dimensional construction made with those simple plus, minus, minus, plus sticks. And when you make it in real, this is the result. This is a sphere that you can build with, in this case, 24 equal elements. And you don't need glue or rope, it just holds itself together. And it has nothing to do with tension, but only with position. There's no part that can fall. Every part is prevented to fall by at least two others. So the total construction will stay as a whole construction. When I found this way of working with the Leonardo grids, I made bigger spheres like this. This is 90 elements in total, well more than one meter in diameter. And to show that this is just a loose construction, I made this small animation. So now when I take away just one element, just for help. So it takes about two hours to build the sphere in two seconds. The spherical concessions also give possibility to use the space inside the sphere, as you can see here. So this element uses the space inside the sphere. So it's not the surface of the sphere that you use, but the complete inside. So this is a real 3D construction. A sphere can be seen as just a surface. So in fact you've got two degrees of freedom. But now you really need the third dimension. And also here, well you recognize the elements now. So with four connecting points here, all over the sphere. But you create a double layer, in fact. And well this is also a nice application of the system. By replacing four of the elements by real fish, you make this one. But also for the fish, four connecting points, that's enough to make the construction stable. You can think of the minimum construction with this system. Well that's this one, the tetraison. Only two elements, but also here the plus, minus, minus plus. So with two elements it just fits. This is the smallest one. After those experiments, well I wanted to create a real big dome. So we replaced the small wooden sticks by real beams. These beams are about four meters long. I was invited to do an installation for an exhibition in Holland. So we built a real big dome. Nice about this kind of dome is that you can build it with just a few people. You don't need much force. That's because you start in the middle and you add sticks at the edge of the concession always. So you only have to lift this part and then you can add another stick or another beam. In total the concession will rise itself and finally you get this one. We built two domes for the exhibition, so two next to each other. When you look from the inside you can see the other one over here. So these were the first domes. At that moment I came in contact with a group called Nexus from Kim Williams. Nexus is a group that studies mathematics and architecture. They were based in Italy and Kim Williams invited me to do a project together with some people of Nexus to study the pages of Leonardo da Vinci a little bit more. That's because on those pages there are a few diagrams, so lines, but there's also a lot of text. That text wasn't translated till that time. So we thought it would be good to really read the text and to see what Leonardo himself says about those diagrams. So we studied the text but also the constructions that are on the page of Leonardo da Vinci. Leonardo da Vinci was telling about the sizes of the beams that he had in mind. We don't know if he really built those domes, but he said it might be handy for a foreign army to make this kind of construction when they are in the forest. They just can take beams and they can put skins over the construction and they have a tent, they can sleep under it, so it could be helpful. So he talked about sizes of the beams and also said something about the height that you get when you use a certain thickness of the beams. So one of the things we did is we really measured if Leonardo was right. We came very close to his measurements. So maybe he really built this. Also tested the strength of the construction, you could walk over it. This was a really nice event. Also at that time I started doing workshops with children. Well, till that time I built domes with just this sticks, but when you build domes with this sticks, they will glide away very easily. So I decided to change the design a little bit and made this type of elements. These are used with this construction. Now you can extend your construction in a very big way. The children are experimenting with the patterns. This was one of the patterns I gave them, but you see when they experiment themselves, they find new patterns. But they follow the rules exactly, so each element is connected in the right way, so no problem. That's what I like with those workshops, that children come with their own ideas, their own patterns. They work together very nicely. They build quite nice constructions. So I had to make a lot of those elements, but it was quite easy to make them. So this is with traditional sticks, and you can see that at a certain moment the construction becomes a little bit steep, and then the sticks will glide away. That's a problem that we also have with the big domes, but I will say something about that later. These are other elements, and here we used a color code. So the mid area were yellow, and you cannot see it, but the end areas we call it green. So they have to connect green to yellow. That's all. That's one simple rule, and they can build a construction. So I went on designing better elements to make better constructions. Well, this was the new design for me, and I made this from plywood. The length is about 40 to 50 centimeters, and I could make them with my milling machine. So I produced a lot of them, and I started to do workshops with those elements. They don't glide away. They connect very easily. Children can build domes in half an hour, big domes, and also the patterns are very clearly to be seen. This was my first experiment with those new elements. So also here you can follow one of the patterns I showed before, but also try to find new patterns. Because you build it so quick, you can experiment in a very easy way. So even with pentagons, heptagons, you can make all kinds of patterns and you can use the system of the Leonardo Grids. In this case, the students thought by themselves, why can't we start with an octagon? So they started with an octagon, and now they had to find out how to go on. They came to the conclusion that you need pentagons and hexagons, and at a certain moment you can go over to only hexagons. But this is what they found themselves. It's also nice to take the complete dome together and tilt it so you have a roof. The construction is very stable and I brought a few elements. So they just lay on top of each other and they can't move anymore. So this works very well and very quickly. This is a workshop that I did in Ghent in Belgium, in Plukas. I was invited by Dirk Heulebruch to do a workshop with his students, students of architecture, and they experimented with the system. You see the different kinds of elements and this was most popular. Here the pattern with the big hexagons, but also here a small hexagons and hexagons in the middle. It's already a combination of two different patterns. In discussions with Dirk Heulebruch, we talked about the possibility of not only having parts of a sphere for your dome, but also trying to make hyperbolic surfaces, so double curved surfaces. After the discussion, I tried to solve this with the same simple elements as we used here. This is the element that we had until then. After some experiments, I came up with another one. Maybe you don't see the difference, but if you put them next to each other, you will see that this one has negative curvature and this one has positive curvature. Now you can use the negative curvature elements at the moment that the construction becomes steep. Then you solve that problem and you can build in another way. You've got another curvature here. You can also build on. You can make the construction bigger and you've got a very wavy surface now. Positive and negative curvature in the same structure. It's fun if you have a lot of elements. This was a project that we did in Portugal, again with the Nexus Group, together with some architects from Portugal. Here we started to build on a hilly surface, but also that doesn't matter. You can just start building your dome and the construction will follow what you're doing. You already see here the positive and the negative curvature. It's very nice when you build it in a group. There's no one who takes the lead. At a certain moment someone decides, well, we need a negative curvature here and then the complete shape of the building is quite natural. As you can see in these few pictures, it just flows from the hill. This is the so-called front view of the building. Here you see that I used two different patterns, so the hexagons and the squares over here. You can combine several patterns in a very good way without losing the system. This is the inside of the construction. Very nice curvature. We did a few more workshops with those elements. This is again in Gantt with the students of architecture. We experimented with several patterns and several shapes of the dome. This is another pattern. Here you can see a top view of one of the domes. A quite natural way of building domes. The system is very suitable to use with children. The last two pictures are with children. A group of children built these themselves. Nice of doing this with small children is that they built something that's bigger than themselves. They can stand in and this is my final picture. Now I will give it over to Enric, who has more experiences with dome building. Okay. First, I am sorry for my English. If you can't understand anything, just tell me. Thanks, Rhinus, for your spectacular work. Thanks to Imaginary for your suggestion to make this talk together with Rhinus. My name is... Okay. What? Okay. Oh. Okay. What? Okay. Okay. Okay. Okay. Okay. Finally, my name is Enric Brasso. I work at the Mathematics Museum of Catalonia, the Maka. Okay. This is a little museum. We have a few financial resources, but we are very active and increasingly recognized. Okay. I talk about my work about this exhibit. Many years ago, we started exhibit the Leonardo Bridge. The Leonardo Bridge is a classical at Science Museum. We find to develop this concept to another dimension. We found and internet this picture and the Rhinus website. But we didn't want to direct copy. We want to make... we have developed my own version. We study the concept. This concept is the reciprocal frame. In the architecture world, it's well known. But in the architecture world, use the different design for each piece. This dom, Leonardo dom, use the same design. Okay. We make this design for this piece. We tested different materials, plastic, cardboard board, and different profiles. Different patterns. We work with this 11. You have seen your work and in your website, Rhinus, you have 100 different patterns. Okay. This is only the simple patterns with some kind of symmetry. This is an infinite structure. We can grow infinitely. But where are the dom? The crucial point is the lower part on the notches, north, aligned. North in the same level. This is change the gap, the different level, and change the curvature. We design for make a big dom with 2011 inside. This is the result. Another question is the notches are not exact. Because I need some freedom of movement. If you have a plan of planning, if the subject of management is not possible to form, and the curvature is possible because there are a freedom of movement. In a moment, it's not possible when the freedom is exhaust, is brick. In this piece, we need some freedom like your piece. This is a very similar design, but difference in the size. We recognize your invention, your work, and we decide to work together for make your... Because this design we put in our website, Freedom, this is PDF, this is our website. We offer to everybody, especially to schools, to make this design for make this activity. We recognize your work, but now we know better your work. My hope is to recognize better this work. This is when the dom progress, the freedom is exhaust, and if the piece can break or slip away, and it's possible to come back like a house of scar, it's collapsed. The construction is fast. Everybody can quickly reproduce the patterns. It involves mental organization for a specific spatial structure in different locations. We work with all age, even with growth, with five years old. The dom is a collaborative work. The combination of the building with the entrance inside the dom is a bright and spectacular ending. We have always tried to have a quiet moment with the people to reflection and question. At the street fairs, the doms are an attraction point. Families and people of all age, all are surprised, whole, with simple patterns, and without fastening elements. A large structure is quickly woody. We share through social network with this hashtag. In conclusion, Leonardo Dom is a rich resource to present mass in different, and varied, and multiple environments. I believe, I have sure, that working together with RINUS, we multiply the dissemination of this activity, and we find new pedagogical possibilities. We invite you to build one or more doms in the space that the team of the conference has prepared. Thank you very much. Thank you very much for your promising presentation. I hope you will find enough helpers. Are there questions? Hello. Thank you. At some point in the initial presentation, there were children who started with an octagon, and eventually found that they have to develop the pattern in a certain way from there. Do they know what the mathematical reason for the pattern changing further on is? That's the first question. My second question is, do people examine which patterns are more robust than others, are stronger than others, and why? And what reasons are these examined by the people who build the dome, or is that discussed afterwards, after the activity or not? To answer the first question, when they solve the problem, when they start with an octagon, and they make a real dome with an octagon in the middle, they do not know the mathematical background, and it may also be a first solution, so maybe there's not a solution as well, but they become now interested in those kinds of problems, and I think that's the important thing. After the dominoes, do they suddenly tell them, okay, this is the fact, the reason why it's happening is that this is the other dominoes. Yeah, well, they start setting up the octagon. That's the first step. And then you've got a few points on the sticks. So for Leonardo's dome pattern, it's important that every notch is used. So when I start here, I have to add one stick here, and that can be in this direction or this direction. So that's the puzzle you have, and then you have to solve it in such a way that every notch is used. So that's it. And the other question, for all the patterns that I made, I also calculated the density. So the number of sticks per square unit, so to say. So you can see that if you have got thousands of sticks and you take this pattern, the dome will be that big or that big. But I'm not that interested in forces. I'm a sculptor. So when I work together with architects, it's their problem to calculate the strength of the construction. Can I add something to that? So the thing is, I work at the Marca Museum with Enrique. I am one of the people who do the workshops. And precisely what you said, the second question, do they realize that there is a difference in size and density? What Reynos was saying, they precisely do, because they build not one dome, they build three, maybe four, because they can build one dome in ten minutes. Our record is three minutes with 30 students and 306, because that's pretty much what you need to add, ten. So the work that you do, the mathematical work, is something that you do after the experiment, not previous to the experiment. And that can be done either immediately or it can be done at school. Usually it is done at school, because this activity can be just translated to planar tiling, but it can also be translated to why don't planar tiling work with a curved structure, which is obviously quite obvious. So talking about 300 and 1000 sticks, how do you organize the mass production and how expensive is it? What we have now here is 1200 sticks and I made them all myself. So... very expensive. The production is a problem, I don't resolve any more, but we study because we hope to expand this and we need more copies in the school. Because this is two euros, three euros for each stick and a multiply is difficult to... What's happened if you remove a piece in the middle, the dome falls down like the sphere? This is a very interesting question. So even when one of the sticks breaks, it will collapse locally, not globally. So that's a very nice property of this construction. We had it with the big domes, with the beams of four meter length. One of the beams, right in the middle, broke. So you can then you think, well, the whole construction will collapse, but no, it's just a local problem and you can solve it locally. Hi. Well, in regard to the production cost, I think I could share our experience later. The fact is we produced some elements with Renault's Contour, the drawing in Taiwan. So in our imaginary exhibition in Taipei, we did the Da Vinci Dome with Renault's. And I also learned that we could probably, in that way, we produced it with the manufacturer, the carpenter. But we could also do it with laser cutters. So we can compare which might be more effective in the cost. Okay, thank you. I'm Helen from Taiwan. Thanks. Yeah, and for schools, you don't have to start with the big ones. You can also start with the smaller ones, as you have seen in the pictures. And I had some. So this is a smaller copy of this one, and it's made by laser cutting, and this maybe costs 10 cents. So you can produce a lot of them and children can experiment and solve mathematical problems with these elements. And well, the fun is, of course, with the big elements, but this is also a possibility. Then I would say we close the inside session and start with the dome building activity. Everybody is invited to join. We'll take place outside, and we will then just float into the lunch break. So whoever is hungry already, please go for lunch, and we'll meet back again here at half past one for the next session.
 Music  In about an hour we are going to build a very big dome outside just near this room.  But before that we will give two different presentations.  One about real experiences with dome building done by Andriik.  And I will start with a small introduction about the history of this complete project.  For me it all started with this object.  Well it does look like a dome I agree.  It's a spherical concession built with just paper.  But this was my first object that led me to the dome concessions.  And this object has a lot to do with the section of a cube.  So I started examining how you could make nice dissections of cubes in different parts.  But at a certain moment I was more concentrated on the lines on the phases of the cubes than the parts itself.  So when you take those lines and you extend them to phases you get an object.  And when you round off all the corners you get this spherical concession that you have seen.  And well I built models in wood.  And well this was the first step for me to come to the dome concessions.  Because when you make the mathematical representation of this object  so you only draw the lines and the connections a graph  then you see that the connecting lines have all the same quality.  They are all connected. Each line is connected to four other lines.  And in a special way because all the endpoints of each of the lines  so the endpoints of this line are connected to one of the midpoints of other lines.  So this is the basic definition of this kind of concession.  And I call this positive, negative, negative, positive.  So all the elements have this property and can be connected in this way.  And I built my first sculpture in this way. This is on a school.  But also here you can see each element is connected with four others.  And endpoints always meet midpoints of other objects.  I found this so interesting that I studied the possibilities of this system a little bit further.  And I came to the conclusion that if you change positive and negative to over and under  like I did here in this drawing, that in fact this is a description of a 3D concession.  A small dome as you can see.  And when you extend the drawing with more elements, well, you'll get a bigger dome.  And finally this was my first dome. Well, you can see it's created in, well, about 25 years ago.  So this is a very old picture.  But for me this was the start to dome building.  So starting from Svirs, I stepped a little bit further.  And studying the system of positive, negative, negative, positive,  it led to a lot of different patterns that I could use to build domes.  And all kinds of domes could be created with the same elements.  The same very simple wooden elements on which I made two incisions  to have the positive, negative, negative, positive structure.  A variety of different patterns, regular but also more irregular.  Here with three pentacons in the middle and, well, the possibilities grow.  And well, that was the moment that I thought, well, this system is so nice and so simple  that I couldn't imagine I was the first one to use this to make this.  It was 1990, so at that time reciprocal structures weren't very well known.  Also internet wasn't available for me to search for other examples.  So I had to go to libraries and I finally found this page  on which I could see drawings that are, well, almost the same as I made.  And this page is a page from a notebook of Leonardo da Vinci.  So a very old source and, well, I had to admit, well, when you compare his drawings to mine,  well, they're almost the same but he was the first.  And also this one, well, this was my drawing, but yeah, I have to admit,  he was the inventor of this system.  Well, I explored the possibilities of these kinds of elements with the connections as I told you  and I was able to make a lot of different patterns.  You see six patterns here, different patterns with all the same properties,  but in total I've got more than 100 different patterns.  Well, I'm from Holland, like M.G. Escher, so I explored some more possibilities of these patterns  and I think this comes close to Escher.  So when I discovered the variety of possibilities with the simple sticks or the simple elements,  the simple positive, negative, negative, positive elements,  I went on searching for other things and one of the things is that you can distort a pattern  without leaving the system because, again, every element is connected in the right way.  So you can make nice distortions and, well, I can show you an animation of such a distortion.  You start with a pattern like this and let's say you take a few elements in the middle  and you just rotate them a little bit.  Then you end up with a distortion but the system is followed in a perfect way  and in real, the distortion in the pattern looks like this.  Well, when you can make a distortion at one point, you can do it all over.  So you can change from one pattern to another in this way.  There are more transformations I discovered but I will leave it by this one.  So from one pattern you can create a new one.  Well, I talked about Escher, so for me it was very simple  to take the step to tiles.  So this line is an element of a dome but it's also a line in between the tiles in this case  but the pattern is exactly the same.  But when you take the step to tiles, you see other possibilities  and one of them is this one.  I can change the tiles a little bit and now I can transform the pattern  from left turning to right turning.  And there's much more that you can do now because when you look at the line  that's in between those tiles, well, you can say this is a simple line  but you can also say this line is the edge of this tile and also the edge of this tile  and the edge of this tile and the edge of this tile.  So in fact this line consists of four different lines, well, like this.  And because of the system, these lines are always equal length  and also these lines are also of equal length.  So in fact you've got a parallelogram and that makes it possible to add some movement to the system like this.  Just because you use the positive, negative, negative, positive element in your pattern.  And well, this has a lot of possibilities.  This is another pattern of the grid, the grid that can be used for a dome  but also be transformed in tiles and then it still works just because of the simple element.  And you can even go further.  When I've got a point with a hexagon here, I can move the bars a little bit away from the center.  So I got space for new smaller bars to make a smaller hexagon, so a fractal step.  And when you translate it into tiles, well, again, the transformation still works.  So this gives a lot of dynamic systems with simple tiles  all based on the patterns with the plus, minus, minus, plus structure.  So here's the last one. So this is, you can see this line is a line that you can use for a dome,  but with the tiles I can make this transformation.  And you can see it transforms into another grid that also has the same quality for the domes.  So with the four points.  So this was one field of exploration for me.  And another field based on this pattern was, well, when you superimpose another grid on the first grid,  you can use the blue grid now to make a polyhedron.  Those triangles can be seen as phases of a polyhedron,  but now this grid is the decoration of the final polyhedron.  So when I fold it together, let's say to an octahedron,  I have the grid completely around this octahedron.  So a three dimensional construction made with those simple plus, minus, minus, plus sticks.  And when you make it in real, this is the result.  This is a sphere that you can build with, in this case, 24 equal elements.  And you don't need glue or rope, it just holds itself together.  And it has nothing to do with tension, but only with position.  There's no part that can fall.  Every part is prevented to fall by at least two others.  So the total construction will stay as a whole construction.  When I found this way of working with the Leonardo grids, I made bigger spheres like this.  This is 90 elements in total, well more than one meter in diameter.  And to show that this is just a loose construction, I made this small animation.  So now when I take away just one element, just for help.  So it takes about two hours to build the sphere in two seconds.  The spherical concessions also give possibility to use the space inside the sphere, as you can see here.  So this element uses the space inside the sphere.  So it's not the surface of the sphere that you use, but the complete inside.  So this is a real 3D construction.  A sphere can be seen as just a surface.  So in fact you've got two degrees of freedom.  But now you really need the third dimension.  And also here, well you recognize the elements now.  So with four connecting points here, all over the sphere.  But you create a double layer, in fact.  And well this is also a nice application of the system.  By replacing four of the elements by real fish, you make this one.  But also for the fish, four connecting points, that's enough to make the construction stable.  You can think of the minimum construction with this system.  Well that's this one, the tetraison.  Only two elements, but also here the plus, minus, minus plus.  So with two elements it just fits.  This is the smallest one.  After those experiments, well I wanted to create a real big dome.  So we replaced the small wooden sticks by real beams.  These beams are about four meters long.  I was invited to do an installation for an exhibition in Holland.  So we built a real big dome.  Nice about this kind of dome is that you can build it with just a few people.  You don't need much force.  That's because you start in the middle and you add sticks at the edge of the concession always.  So you only have to lift this part and then you can add another stick or another beam.  In total the concession will rise itself and finally you get this one.  We built two domes for the exhibition, so two next to each other.  When you look from the inside you can see the other one over here.  So these were the first domes.  At that moment I came in contact with a group called Nexus from Kim Williams.  Nexus is a group that studies mathematics and architecture.  They were based in Italy and Kim Williams invited me to do a project together with some people of Nexus  to study the pages of Leonardo da Vinci a little bit more.  That's because on those pages there are a few diagrams, so lines, but there's also a lot of text.  That text wasn't translated till that time.  So we thought it would be good to really read the text and to see what Leonardo himself says about those diagrams.  So we studied the text but also the constructions that are on the page of Leonardo da Vinci.  Leonardo da Vinci was telling about the sizes of the beams that he had in mind.  We don't know if he really built those domes, but he said it might be handy for a foreign army  to make this kind of construction when they are in the forest.  They just can take beams and they can put skins over the construction and they have a tent, they can sleep under it,  so it could be helpful.  So he talked about sizes of the beams and also said something about the height that you get when you use a certain thickness of the beams.  So one of the things we did is we really measured if Leonardo was right.  We came very close to his measurements.  So maybe he really built this.  Also tested the strength of the construction, you could walk over it.  This was a really nice event.  Also at that time I started doing workshops with children.  Well, till that time I built domes with just this sticks,  but when you build domes with this sticks, they will glide away very easily.  So I decided to change the design a little bit and made this type of elements.  These are used with this construction.  Now you can extend your construction in a very big way.  The children are experimenting with the patterns.  This was one of the patterns I gave them, but you see when they experiment themselves, they find new patterns.  But they follow the rules exactly, so each element is connected in the right way, so no problem.  That's what I like with those workshops, that children come with their own ideas, their own patterns.  They work together very nicely.  They build quite nice constructions.  So I had to make a lot of those elements, but it was quite easy to make them.  So this is with traditional sticks, and you can see that at a certain moment the construction becomes a little bit steep,  and then the sticks will glide away.  That's a problem that we also have with the big domes, but I will say something about that later.  These are other elements, and here we used a color code.  So the mid area were yellow, and you cannot see it, but the end areas we call it green.  So they have to connect green to yellow.  That's all. That's one simple rule, and they can build a construction.  So I went on designing better elements to make better constructions.  Well, this was the new design for me, and I made this from plywood.  The length is about 40 to 50 centimeters, and I could make them with my milling machine.  So I produced a lot of them, and I started to do workshops with those elements.  They don't glide away. They connect very easily.  Children can build domes in half an hour, big domes, and also the patterns are very clearly to be seen.  This was my first experiment with those new elements.  So also here you can follow one of the patterns I showed before, but also try to find new patterns.  Because you build it so quick, you can experiment in a very easy way.  So even with pentagons, heptagons, you can make all kinds of patterns and you can use the system of the Leonardo Grids.  In this case, the students thought by themselves, why can't we start with an octagon?  So they started with an octagon, and now they had to find out how to go on.  They came to the conclusion that you need pentagons and hexagons, and at a certain moment you can go over to only hexagons.  But this is what they found themselves.  It's also nice to take the complete dome together and tilt it so you have a roof.  The construction is very stable and I brought a few elements.  So they just lay on top of each other and they can't move anymore.  So this works very well and very quickly.  This is a workshop that I did in Ghent in Belgium, in Plukas.  I was invited by Dirk Heulebruch to do a workshop with his students, students of architecture, and they experimented with the system.  You see the different kinds of elements and this was most popular.  Here the pattern with the big hexagons, but also here a small hexagons and hexagons in the middle.  It's already a combination of two different patterns.  In discussions with Dirk Heulebruch, we talked about the possibility of not only having parts of a sphere for your dome,  but also trying to make hyperbolic surfaces, so double curved surfaces.  After the discussion, I tried to solve this with the same simple elements as we used here.  This is the element that we had until then.  After some experiments, I came up with another one.  Maybe you don't see the difference, but if you put them next to each other, you will see that this one has negative curvature and this one has positive curvature.  Now you can use the negative curvature elements at the moment that the construction becomes steep.  Then you solve that problem and you can build in another way.  You've got another curvature here.  You can also build on.  You can make the construction bigger and you've got a very wavy surface now.  Positive and negative curvature in the same structure.  It's fun if you have a lot of elements.  This was a project that we did in Portugal, again with the Nexus Group, together with some architects from Portugal.  Here we started to build on a hilly surface, but also that doesn't matter.  You can just start building your dome and the construction will follow what you're doing.  You already see here the positive and the negative curvature.  It's very nice when you build it in a group.  There's no one who takes the lead.  At a certain moment someone decides, well, we need a negative curvature here and then the complete shape of the building is quite natural.  As you can see in these few pictures, it just flows from the hill.  This is the so-called front view of the building.  Here you see that I used two different patterns, so the hexagons and the squares over here.  You can combine several patterns in a very good way without losing the system.  This is the inside of the construction.  Very nice curvature.  We did a few more workshops with those elements.  This is again in Gantt with the students of architecture.  We experimented with several patterns and several shapes of the dome.  This is another pattern.  Here you can see a top view of one of the domes.  A quite natural way of building domes.  The system is very suitable to use with children.  The last two pictures are with children.  A group of children built these themselves.  Nice of doing this with small children is that they built something that's bigger than themselves.  They can stand in and this is my final picture.  Now I will give it over to Enric, who has more experiences with dome building.  Okay.  First, I am sorry for my English.  If you can't understand anything, just tell me.  Thanks, Rhinus, for your spectacular work.  Thanks to Imaginary for your suggestion to make this talk together with Rhinus.  My name is...  Okay.  What?  Okay.  Oh.  Okay.  What?  Okay.  Okay.  Okay.  Okay.  Okay.  Finally, my name is Enric Brasso.  I work at the Mathematics Museum of Catalonia, the Maka.  Okay.  This is a little museum.  We have a few financial resources, but we are very active and increasingly recognized.  Okay.  I talk about my work about this exhibit.  Many years ago, we started exhibit the Leonardo Bridge.  The Leonardo Bridge is a classical at Science Museum.  We find to develop this concept to another dimension.  We found and internet this picture and the Rhinus website.  But we didn't want to direct copy.  We want to make... we have developed my own version.  We study the concept.  This concept is the reciprocal frame.  In the architecture world, it's well known.  But in the architecture world, use the different design for each piece.  This dom, Leonardo dom, use the same design.  Okay.  We make this design for this piece.  We tested different materials, plastic, cardboard board, and different profiles.  Different patterns.  We work with this 11.  You have seen your work and in your website, Rhinus, you have 100 different patterns.  Okay.  This is only the simple patterns with some kind of symmetry.  This is an infinite structure.  We can grow infinitely.  But where are the dom?  The crucial point is the lower part on the notches, north, aligned.  North in the same level.  This is change the gap, the different level, and change the curvature.  We design for make a big dom with 2011 inside.  This is the result.  Another question is the notches are not exact.  Because I need some freedom of movement.  If you have a plan of planning,  if the subject of management is not possible to form,  and the curvature is possible because there are a freedom of movement.  In a moment, it's not possible when the freedom is exhaust, is brick.  In this piece, we need some freedom like your piece.  This is a very similar design, but difference in the size.  We recognize your invention, your work, and we decide to work together for make your...  Because this design we put in our website, Freedom, this is PDF, this is our website.  We offer to everybody, especially to schools,  to make this design for make this activity.  We recognize your work, but now we know better your work.  My hope is to recognize better this work.  This is when the dom progress, the freedom is exhaust,  and if the piece can break or slip away,  and it's possible to come back like a house of scar, it's collapsed.  The construction is fast. Everybody can quickly reproduce the patterns.  It involves mental organization for a specific spatial structure in different locations.  We work with all age, even with growth, with five years old.  The dom is a collaborative work.  The combination of the building with the entrance inside the dom is a bright and spectacular ending.  We have always tried to have a quiet moment with the people to reflection and question.  At the street fairs, the doms are an attraction point.  Families and people of all age, all are surprised, whole, with simple patterns,  and without fastening elements.  A large structure is quickly woody. We share through social network with this hashtag.  In conclusion, Leonardo Dom is a rich resource to present mass in different,  and varied, and multiple environments. I believe, I have sure, that working together with RINUS,  we multiply the dissemination of this activity, and we find new pedagogical possibilities.  We invite you to build one or more doms in the space that the team of the conference has prepared.  Thank you very much.  Thank you very much for your promising presentation.  I hope you will find enough helpers.  Are there questions?  Hello. Thank you.  At some point in the initial presentation, there were children who started with an octagon, and eventually found that they have to develop the pattern in a certain way from there.  Do they know what the mathematical reason for the pattern changing further on is?  That's the first question.  My second question is, do people examine which patterns are more robust than others, are stronger than others, and why?  And what reasons are these examined by the people who build the dome, or is that discussed afterwards, after the activity or not?  To answer the first question, when they solve the problem, when they start with an octagon, and they make a real dome with an octagon in the middle,  they do not know the mathematical background, and it may also be a first solution, so maybe there's not a solution as well,  but they become now interested in those kinds of problems, and I think that's the important thing.  After the dominoes, do they suddenly tell them, okay, this is the fact, the reason why it's happening is that this is the other dominoes.  Yeah, well, they start setting up the octagon. That's the first step.  And then you've got a few points on the sticks.  So for Leonardo's dome pattern, it's important that every notch is used.  So when I start here, I have to add one stick here, and that can be in this direction or this direction.  So that's the puzzle you have, and then you have to solve it in such a way that every notch is used.  So that's it.  And the other question, for all the patterns that I made, I also calculated the density.  So the number of sticks per square unit, so to say.  So you can see that if you have got thousands of sticks and you take this pattern, the dome will be that big or that big.  But I'm not that interested in forces. I'm a sculptor.  So when I work together with architects, it's their problem to calculate the strength of the construction.  Can I add something to that? So the thing is, I work at the Marca Museum with Enrique.  I am one of the people who do the workshops.  And precisely what you said, the second question, do they realize that there is a difference in size and density?  What Reynos was saying, they precisely do, because they build not one dome, they build three, maybe four, because they can build one dome in ten minutes.  Our record is three minutes with 30 students and 306, because that's pretty much what you need to add, ten.  So the work that you do, the mathematical work, is something that you do after the experiment, not previous to the experiment.  And that can be done either immediately or it can be done at school.  Usually it is done at school, because this activity can be just translated to planar tiling,  but it can also be translated to why don't planar tiling work with a curved structure, which is obviously quite obvious.  So talking about 300 and 1000 sticks, how do you organize the mass production and how expensive is it?  What we have now here is 1200 sticks and I made them all myself.  So... very expensive.  The production is a problem, I don't resolve any more, but we study because we hope to expand this and we need more copies in the school.  Because this is two euros, three euros for each stick and a multiply is difficult to...  What's happened if you remove a piece in the middle, the dome falls down like the sphere?  This is a very interesting question.  So even when one of the sticks breaks, it will collapse locally, not globally.  So that's a very nice property of this construction.  We had it with the big domes, with the beams of four meter length.  One of the beams, right in the middle, broke.  So you can then you think, well, the whole construction will collapse, but no, it's just a local problem and you can solve it locally.  Hi. Well, in regard to the production cost, I think I could share our experience later.  The fact is we produced some elements with Renault's Contour, the drawing in Taiwan.  So in our imaginary exhibition in Taipei, we did the Da Vinci Dome with Renault's.  And I also learned that we could probably, in that way, we produced it with the manufacturer, the carpenter.  But we could also do it with laser cutters.  So we can compare which might be more effective in the cost.  Okay, thank you. I'm Helen from Taiwan. Thanks.  Yeah, and for schools, you don't have to start with the big ones.  You can also start with the smaller ones, as you have seen in the pictures.  And I had some.  So this is a smaller copy of this one, and it's made by laser cutting, and this maybe costs 10 cents.  So you can produce a lot of them and children can experiment and solve mathematical problems with these elements.  And well, the fun is, of course, with the big elements, but this is also a possibility.  Then I would say we close the inside session and start with the dome building activity.  Everybody is invited to join.  We'll take place outside, and we will then just float into the lunch break.  So whoever is hungry already, please go for lunch, and we'll meet back again here at half past one for the next session.
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