Return to Question

According to wikipedia, "The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges." And according to this question, the steps to create a chamfered polyhedron are as follows:

1. shrink the given polyhedron's faces by a factor $r, 0<r<1$
2. translate the shrunken faces outward [I'll call this distance $s$], so that their edges are coplanar with the original polyhedron's edges
3. construct the required hexagons from the original polyhedron's vertices and the vertices of the shrunk faces.

I haven't found any other explicit details regarding the relationship between $r$ and the shape of the hexagons, and so my questions are these (note that the value of $s$ will be directly dependent on the shape of the original face, and so $s$ will need to be represented with some parameter(s) $x$ that represent the face's shape somehow):

1. What is the general relationship between $r$ and $s$ in order to get coplanar vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
2. What is the general relationship between $r$ and $s$ in order to get equilateral vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
3. For equilateral and coplanar vertices, what value does $r$ need to be so that the diameter of the resulting hexagon equals the original side length? (ie. given pentagons that share edge $AB$, the resulting hexagon between the two would have a diameter equal to the length of $AB$ - shown in the images below) I assume this relationship could be obtained by setting up the answers to the previous questions as a system of equations

Note that this is different than my question here as it takes a different approach in constructing the chamfered faces - scaling along the relative Z axis vs. translating.

According to wikipedia, "The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges." And according to this question, the steps to create a chamfered polyhedron are as follows:

1. shrink the given polyhedron's faces by a factor $r, 0<r<1$
2. translate the shrunken faces outward [I'll call this distance $s$], so that their edges are coplanar with the original polyhedron's edges
3. construct the required hexagons from the original polyhedron's vertices and the vertices of the shrunk faces.

I haven't found any other explicit details regarding the relationship between $r$ and the shape of the hexagons, and so my questions are these (note that the value of $s$ will be directly dependent on the shape of the original face, and so $s$ will need to be represented with some parameter(s) that represent the face's shape somehow):

1. What is the general relationship between $r$ and $s$ in order to get coplanar vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
2. What is the general relationship between $r$ and $s$ in order to get equilateral vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
3. For equilateral and coplanar vertices, what value does $r$ need to be so that the diameter of the resulting hexagon equals the original side length? (ie. given pentagons that share edge $AB$, the resulting hexagon between the two would have a diameter equal to the length of $AB$ - shown in the images below) I assume this relationship could be obtained by setting up the answers to the previous questions as a system of equations

Note that this is different than my question here as it takes a different approach in constructing the chamfered faces - scaling along the relative Z axis vs. translating.

According to [wikipedia][1], "The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges." And according to [this question][2], the steps to create a chamfered polyhedron are as follows:

1. shrink the given polyhedron's faces by a factor $r, 0<r<1$
2. translate the shrunken faces outward [I'll call this distance $s$], so that their edges are coplanar with the original polyhedron's edges
3. construct the required hexagons from the original polyhedron's vertices and the vertices of the shrunk faces.

I haven't found any other explicit details regarding the relationship between $r$ and the shape of the hexagons, and so my questions are these (note that the value of $s$ will be directly dependent on the shape of the original face, and so $s$ will need to be represented with some parameter(s) $x$ that represent the face's shape somehow):

1. What is the general relationship between $r$ and $s$ in order to get coplanar vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
2. What is the general relationship between $r$ and $s$ in order to get equilateral vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
3. For equilateral and coplanar vertices, what value does $r$ need to be so that the diameter of the resulting hexagon equals the original side length? (ie. given pentagons that share edge $AB$, the resulting hexagon between the two would have a diameter equal to the length of $AB$ - shown in the images below) I assume this relationship could be obtained by setting up the answers to the previous questions as a system of equations

[![enter image description here][3]][3][![enter image description here][4]][4]

Note that this is different than my question [here][5] as it takes a different approach in constructing the chamfered faces - scaling along the relative Z axis vs. translating.

According to wikipedia, "The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges." And according to this question, the steps to create a chamfered polyhedron are as follows:

1. shrink the given polyhedron's faces by a factor $r, 0<r<1$
2. translate the shrunken faces outward [I'll call this distance $s$], so that their edges are coplanar with the original polyhedron's edges
3. construct the required hexagons from the original polyhedron's vertices and the vertices of the shrunk faces.

I haven't found any other explicit details regarding the relationship between $r$ and the shape of the hexagons, and so my questions are these (note that the value of $s$ will be directly dependent on the shape of the original face, and so $s$ will need to be represented with some parameter(s) $x$ that represent the face's shape somehow):

1. What is the general relationship between $r$ and $s$ in order to get coplanar vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
2. What is the general relationship between $r$ and $s$ in order to get equilateral vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
3. For equilateral and coplanar vertices, what value does $r$ need to be so that the diameter of the resulting hexagon equals the original side length? (ie. given pentagons that share edge $AB$, the resulting hexagon between the two would have a diameter equal to the length of $AB$ - shown in the images below) I assume this relationship could be obtained by setting up the answers to the previous questions as a system of equations

Note that this is different than my question here as it takes a different approach in constructing the chamfered faces - scaling along the relative Z axis vs. translating.

According to wikipedia, "The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges." And according to this question, the steps to create a chamfered polyhedron are as follows:

1. shrink the given polyhedron's faces by a factor $r, 0<r<1$
2. translate the shrunken faces outward [I'll call this distance $s$], so that their edges are coplanar with the original polyhedron's edges
3. construct the required hexagons from the original polyhedron's vertices and the vertices of the shrunk faces.

I haven't found any other explicit details regarding the relationship between $r$ and the shape of the hexagons, and so my questions are these (note that the value of $s$ will be directly dependent on the shape of the original face, and so $s$ will need to be represented with some parameter(s) $x$ that represent the face's shape somehow):

1. What is the general relationship between $r$ and $s$ in order to get coplanar vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
2. What is the general relationship between $r$ and $s$ in order to get equilateral vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
3. For equilateral and coplanar vertices, what value does $r$ need to be so that the diameter of the resulting hexagon equals the original side length? (ie. given pentagons that share edge $AB$, the resulting hexagon between the two would have a diameter equal to the length of $AB$ - shown in the images below) I assume this relationship could be obtained by setting up the answers to the previous questions as a system of equations

Note that this is different than my question here as it takes a different approach in constructing the chamfered faces - scaling along the relative Z axis vs. translating.

According to [wikipedia][1], "The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges." And according to [this question][2], the steps to create a chamfered polyhedron are as follows:

1. shrink the given polyhedron's faces by a factor $r, 0<r<1$
2. translate the shrunken faces outward [I'll call this distance $s$], so that their edges are coplanar with the original polyhedron's edges
3. construct the required hexagons from the original polyhedron's vertices and the vertices of the shrunk faces.

I haven't found any other explicit details regarding the relationship between $r$ and the shape of the hexagons, and so my questions are these (note that the value of $s$ will be directly dependent on the shape of the original face, and so $s$ will need to be represented with some parameter(s) $x$ that represent the face's shape somehow):

1. What is the general relationship between $r$ and $s$ in order to get coplanar vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
2. What is the general relationship between $r$ and $s$ in order to get equilateral vertices? Does this relationship change depending on how many times the polyhedron is chamfered?
3. For equilateral and coplanar vertices, what value does $r$ need to be so that the diameter of the resulting hexagon equals the original side length? (ie. given pentagons that share edge $AB$, the resulting hexagon between the two would have a diameter equal to the length of $AB$ - shown in the images below) I assume this relationship could be obtained by setting up the answers to the previous questions as a system of equations

[![enter image description here][3]][3][![enter image description here][4]][4]

Note that this is different than my question [here][5] as it takes a different approach in constructing the chamfered faces - scaling along the relative Z axis vs. translating.