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The family CSM
1/ Generalities
1.1/ The origin : A module composed of modules.
Here is the first family of closed forms. By this, I mean forms closed within themselves. This idea came to me when I first noticed that family C6 forms were sufficiently flexible to adopt certain proportions allowing them to surround a dodecahedron.
This family is thus the first of my families (there are others) to establish a connection with platonic volumes. To illustrate the explanation, I used the form 36P58P2.C6, but I could have used form 36P58P3 or P4 or P5, etc. (See C6 family).
This form is interesting in this particular case because its five suspension points can rest on each vertex of the pentagons that make up the dodecahedron. I calculated the slope of the angle at 58.2825255885° which is (as any geodesic-man knows..) the half angle that makes up the faces of the dodecahedron. (Module and Joining)
The shapes are then placed on each faces of the dodecahedron. As their openings line up joining perfectly, a closed volume with three orthogonally symmetric planes is formed. I call this shape "the mother form". Each tip of this form is inscribed in a sphere. By cutting the mother form on each of its three symmetrical planes, I get three identical forms. (the classic sights and the mother shape)
1.2/ Diversification
After having cut the mother form by its symmetrical plane, the shape lies completely on a plane. As illustrated, (Mother shape n°11 and 21) equilateral half triangles which result from the split appear at certain points of the base.
If we consider each vertex as being articulated, the half form is not rigid in itself, but thanks to the miracle of the god of structures, we easily understand when the base line is rigid, either because it is fixed on the plane (on the ground) or by any other means of rigidifying the line, the entire form becomes completely inflexible. Of course, by this I mean geometrically rigid (which is already considerable).
The materials, being pliable, do not behave (alas, or so much the better!) like perfect straight lines in geometry.
When carefully examining mother shape n°22, one notices (arrow) that the broken line directly under the one on the ground is not quite horizontal. From this observation, I got the idea that by removing the first row of triangles, it is possible to hang this line towards the horizontal one. The equilateral distribution must, of course, be kept. It was possible…a true miracle! All the illustrations which follow are examples of this miracle repeated.
1.3/Architectural remarks
Among all of the families studied, notice that forms stemming from the CSM family are those that are most similar to the sphere; particularly the mother forms whose tips are inscribed on the surface of the sphere. Of course, this is only a geometrical consequence and not a sought-after goal.
Only in a few cases are there objective reasons for insisting upon using the sphere. Although the circle, cylinder or sphere were used in Romanesque architecture, it was for geometrical simplicity rather than having a specific intent linked to these forms.
When men found other possibilities like the ogive (diagonal rib), Gothic architecture evolved and older techniques were abandoned. Lightened structures, achievements in stability and especially the economic savings resulting from this choice were determining factors in the use of this new construction technique.
Thus, because men are destined to adapt or perish, their fate orders them to distinguish the path that leads them to success or to failure. Sometimes experience allows us to anticipate the timeliness of one or the other until the events themselves contradict the most tried out experience. And prudence that commands "to wait and see" is in itself defeated by the most daring. Perhaps between boldness and cautiousness…! It is better to laugh !
1.1/ The origin : A module composed of modules.
Here is the first family of closed forms. By this, I mean forms closed within themselves. This idea came to me when I first noticed that family C6 forms were sufficiently flexible to adopt certain proportions allowing them to surround a dodecahedron.
This family is thus the first of my families (there are others) to establish a connection with platonic volumes. To illustrate the explanation, I used the form 36P58P2.C6, but I could have used form 36P58P3 or P4 or P5, etc. (See C6 family).
This form is interesting in this particular case because its five suspension points can rest on each vertex of the pentagons that make up the dodecahedron. I calculated the slope of the angle at 58.2825255885° which is (as any geodesic-man knows..) the half angle that makes up the faces of the dodecahedron. (Module and Joining)
The shapes are then placed on each faces of the dodecahedron. As their openings line up joining perfectly, a closed volume with three orthogonally symmetric planes is formed. I call this shape "the mother form". Each tip of this form is inscribed in a sphere. By cutting the mother form on each of its three symmetrical planes, I get three identical forms. (the classic sights and the mother shape)
1.2/ Diversification
After having cut the mother form by its symmetrical plane, the shape lies completely on a plane. As illustrated, (Mother shape n°11 and 21) equilateral half triangles which result from the split appear at certain points of the base.
If we consider each vertex as being articulated, the half form is not rigid in itself, but thanks to the miracle of the god of structures, we easily understand when the base line is rigid, either because it is fixed on the plane (on the ground) or by any other means of rigidifying the line, the entire form becomes completely inflexible. Of course, by this I mean geometrically rigid (which is already considerable).
The materials, being pliable, do not behave (alas, or so much the better!) like perfect straight lines in geometry.
When carefully examining mother shape n°22, one notices (arrow) that the broken line directly under the one on the ground is not quite horizontal. From this observation, I got the idea that by removing the first row of triangles, it is possible to hang this line towards the horizontal one. The equilateral distribution must, of course, be kept. It was possible…a true miracle! All the illustrations which follow are examples of this miracle repeated.
1.3/Architectural remarks
Among all of the families studied, notice that forms stemming from the CSM family are those that are most similar to the sphere; particularly the mother forms whose tips are inscribed on the surface of the sphere. Of course, this is only a geometrical consequence and not a sought-after goal.
Only in a few cases are there objective reasons for insisting upon using the sphere. Although the circle, cylinder or sphere were used in Romanesque architecture, it was for geometrical simplicity rather than having a specific intent linked to these forms.
When men found other possibilities like the ogive (diagonal rib), Gothic architecture evolved and older techniques were abandoned. Lightened structures, achievements in stability and especially the economic savings resulting from this choice were determining factors in the use of this new construction technique.
Thus, because men are destined to adapt or perish, their fate orders them to distinguish the path that leads them to success or to failure. Sometimes experience allows us to anticipate the timeliness of one or the other until the events themselves contradict the most tried out experience. And prudence that commands "to wait and see" is in itself defeated by the most daring. Perhaps between boldness and cautiousness…! It is better to laugh !