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Polyhedra with articulated faces |
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Polyhedron
A polyhedron can be defined in many ways. For instance, polyhedra have often been defined as solids. However,
they have also been defined as a set of rigid surfaces linked at their edges by hinges. The resulting construction
is generally rigid, with some very rare exceptions. Polyhedra can also be defined as frameworks, which are obtained
by placing bars at the edges, the bars being connected at the vertices by spherical joints. If all the faces are
triangular, the framework is generally rigid. In the other cases, the framework is mobile and loses its initial
polyhedral shape once deformed. General information on polyhedra can be found at
Wikipedia, at
The Pavilion of Polyhedreality and at
Wolfram MathWorld.
Fig. 1: Solid dodecahedron and cubic framework.
Definition
Here, a new definition is proposed. Consider a polyhedron as a framework in which the faces are constrained to
remain planar. The resulting construction is referred to here as polyhedron with articulated faces (PAF).
In some cases the PAFs are rigid structures, while in others they are articulated mechanisms, which possess
interesting kinematic properties.
Construction
Implementing the planarity constraint mechanically in a framework would be difficult.
Instead, it is proposed here to proceed from the outset with a new mechanical construction, which is covered
by US patent (No. 7,118,442).
First, the faces are built as closed-loop planar linkages using a set of links forming the sides of the
polygons. These links are connected by revolute joints at the corners, the axes of the joints being perpendicular
to the plane formed by the face.
This ensures the planarity of the faces for any configuration. Then, the
sides of the faces are connected by revolute joints that lie on the edges of the polyhedron and intersect
the joints of the faces at the corner of the faces.
Therefore, all the joints associated to a given vertex of the polyhedron intersect at the vertex for any configuration.
It is pointed out that the construction of PAFs only require one type of part. Moreover, for PAFs whose
edges all have the same length, all the parts are identical.
Fig. 2: Construction of a cubic PAF: part, polygon and two configurations of the polyhedron.
Mobility
The main interest of the PAFs is their mobility. Some of them are rigid structures while others are articulated
mechanisms that deform with nice kinematic properties. Also, some of them are locally mobile but globally rigid.
In other words, they can only move in their initial configuration. A kinematic chain with such properties, which is
relatively rare, is often called shaky. In order to determine their mobility, i.e. how many degrees of
freedom they have, a general method, which involves the first derivative of the constraints equations, is developed.
Also, numerical simulations are performed in order to observe the flexed configurations or to find which are shaky PAFs.
Finally, plastic models are built. Because of the flexibility of the plastic and the clearance of the joints, the plastic
models are more flexible than they should be in theory. As a result, this allows to observe the flexing of the shaky PAFs,
which do not significantly move in simulations. It is noted that the range of motion of the plastic models is limited
by mechanical interference between adjacent parts. More details on PAFs, particularly on the study of their mobility,
can be found in the paper
Polyhedra with Articulated Faces, from IFToMM 2007 congress.
Photos and videos
Here, the PAFs generated from the five Platonic solids (or regular polyhedra) and the thirteen Archimedean solids
(or semiregular polyhedra) are illustrated and their mobility is given. The photos of plastic models can be enlarged
by clicking on the image.
Platonic solids
- Tetrahedron (4 faces, 6 edges, 4 vertices): rigid
- Cube (6 faces, 12 edges, 8 vertices): mobile (3 DOFS)
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PAF of a Cube
Video clip showing the mobility of a PAF made of a cube.
Format: mpg Length: 0:18 Size: 3.2 Mb
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- Octahedron (8 faces, 12 edges, 6 vertices): rigid
- Dodecahedron (12 faces, 30 edges, 20 vertices): shaky (5 DOFS)
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PAF of a Dodecahedron
Video clip showing the shaky mobility of a PAF made of a dodecahedron.
Format: mpg Length: 0:20 Size: 3.7 Mb
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- Icosahedron (20 faces, 30 edges, 12 vertices): rigid
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PAF of an Icosahedron
Video clip showing the rigidity of a PAF made of an icosahedron.
Format: mpg Length: 0:12 Size: 2.3 Mb
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Archimedean solids
- Truncated tetrahedron (8 faces, 18 edges, 12 vertices): rigid
- Cuboctahedron (14 faces, 24 edges, 12 vertices): mobile (3 DOFS)
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PAF of a Cuboctahedron
Video clip showing the mobility of a PAF made of a cuboctahedron.
Format: mpg Length: 0:17 Size: 3.1 Mb
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- Truncated octahedron (14 faces, 36 edges, 24 vertices): mobile (5 DOFS)
- Truncated cube (14 faces, 36 edges, 24 vertices): rigid
- Rhombicuboctahedron (26 faces, 48 edges, 24 vertices): mobile (6 DOFS)
- Snub cube (38 faces, 60 edges, 24 vertices): rigid
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PAF of a Snub cube
Video clip showing the rigidity of a snub cube.
Format: mpg Length: 0:16 Size: 3.0 Mb
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- Icosidodecahedron (32 faces, 60 edges, 30 vertices): rigid
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PAF of an Icosidodecahedron
Video clip showing the rigidity of a PAF made of an icosidodecahedron.
Format: mpg Length: 0:12 Size: 2.3 Mb
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- Great rhombicuboctahedron (26 faces, 72 edges, 48 vertices): mobile (5 DOFS)
- Truncated icosahedron (32 faces, 90 edges, 60 vertices): shaky (5 DOFS)
- Truncated dodecahedron (32 faces, 90 edges, 60 vertices): shaky (4 DOFS)
- Rhombicosidodecahedron (62 faces, 120 edges, 60 vertices): mobile (3 DOFS) and shaky (5 DOFS)
- Snub dodecahedron (92 faces, 150 edges, 60 vertices): rigid
- Great rhombicosidodecahedron (62 faces, 180 edges, 120 vertices): mobile (5 DOFS)
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