Proper name of this polyhedron?

[Tchmstr]

Hello, I'm hoping to find the most accurate and proper name of this shape, if there is one, or for one to be named. I'm sure I'm overthinking it, but I'm just curious. It's simply 12 joined Icosahedrons, so my first thought was Dodecaicosahedron, but that doesn't seem right. I then thought maybe it's Octacontahecatohedron as there are 180 faces. The more I look at it though, the more I think I may not even be close. As there are some faces that make it no longer completely convex, I'm not sure if that alters the definition or not. Please post your thoughts, thank you.

 

[Dr.Peterson]

How are they "joined"? Just along edges?

In what sense is it a single polyhedron at all? Whatever it is, it certainly is not convex, and it is not just "certain faces" that make it so.

In any case, a polyhedron with 180 faces is most reasonably just called a 180-hedron.

 

[Tchmstr]

How are they "joined"? Just along edges?

In what sense is it a single polyhedron at all? Whatever it is, it certainly is not convex, and it is not just "certain faces" that make it so.

In any case, a polyhedron with 180 faces is most reasonably just called a 180-hedron.

Thank you for your response.

This is just a mock up, I would actually be making this from individual triangles. The simplest way I can explain it is, take a Icosahedron and remove 5 adjoining triangles to reveal an open pentagon, then join on the edge. As to your question of "In what sense is it a single polyhedron at all?', please enlighten me if I am missing something in regards to the definition of a polyhedron, or is there's a more appropriate term for this? Back to naming, the disdyakis triacontahedron is a 120 faced polyhedron, so why could there not be a proper name for more faces?

 

[topsquark]

It would seem that a disdyakis triacontahedron is a regular solid (and convex, if that's the right term) to boot. Your figure is neither so it would technically require a different name. What name that would be is beyond me.

-Dan

 

[Tchmstr]

It would seem that a disdyakis triacontahedron is a regular solid (and convex, if that's the right term) to boot. Your figure is neither so it would technically require a different name. What name that would be is beyond me.

-Dan

Right, that's the part that's hanging me up more than anything, but I'm not sure if it should. A concave polyhedron, is still a polyhedron, at least that's what every text I'm seeing on the matter states. If there is some specific rule that this is breaking for being a polyhedron, I'm simply unaware of it. I have seen somewhat similar shapes still referred to as polyhedrons, but I don't know if that means it is correct.
This brings me back to 'what classifies a mountain as a mountain.'

 

[Dr.Peterson]

Thank you for your response.

This is just a mock up, I would actually be making this from individual triangles. The simplest way I can explain it is, take a Icosahedron and remove 5 adjoining triangles to reveal an open pentagon, then join on the edge. As to your question of "In what sense is it a single polyhedron at all?', please enlighten me if I am missing something in regards to the definition of a polyhedron, or is there's a more appropriate term for this? Back to naming, the disdyakis triacontahedron is a 120 faced polyhedron, so why could there not be a proper name for more faces?

Since you didn't give details about how you are making this, and specifically did not say you would be removing any faces (quote: "It's simply 12 joined Icosahedrons"), I took it as gluing whole icosahedra together along edges. That would not be a polyhedron, because more than two faces would meet on an edge! The number 180 would not fit that, but I took you at your word ...

But there is no standard name for this, unless someone else has done exactly the same thing. So I suppose, if it deserves a name at all, you are free to make one up. (And others would be free to use it or reject it.) That's how names work.

 

[Tchmstr]

Since you didn't give details about how you are making this, and specifically did not say you would be removing any faces (quote: "It's simply 12 joined Icosahedrons"), I took it as gluing whole icosahedra together along edges. That would not be a polyhedron, because more than two faces would meet on an edge! The number 180 would not fit that, but I took you at your word ...

But there is no standard name for this, unless someone else has done exactly the same thing. So I suppose, if it deserves a name at all, you are free to make one up. (And others would be free to use it or reject it.) That's how names work.

You're correct, I did not give enough information initially. I simply used the icosahedra 'glued together' as a base to template the design.
(20 - 5) x 12 =180. Assuming proper construction, not 'glued together', is there any reason it is not a polyhedron? Would it be classified as a regular polyhedron, concave polyhedron, nonconvex, etc.? I'm doing my best to present my questions out of pure interest and curiosity, most of my information on this has come from some wiki or other documentation. If I am going to bother making/naming this thing, I'd rather it either be somewhat proper/descriptive, or just something catchy if it's not a 'real' polyhedron. Again, thank you for your time and thoughts.

 

[Dr.Peterson]

You're correct, I did not give enough information initially. I simply used the icosahedra 'glued together' as a base to template the design.
(20 - 5) x 12 =180. Assuming proper construction, not 'glued together', is there any reason it is not a polyhedron? Would it be classified as a regular polyhedron, concave polyhedron, nonconvex, etc.? I'm doing my best to present my questions out of pure interest and curiosity, most of my information on this has come from some wiki or other documentation. If I am going to bother making/naming this thing, I'd rather it either be somewhat proper/descriptive, or just something catchy if it's not a 'real' polyhedron. Again, thank you for your time and thoughts.

I thought I made it clear that, as now described, it is a polyhedron.

I would hope you know that it is not convex and is not regular, because it (obviously) does not fit either definition.

But you should also know, from whatever you've read about naming polyhedra, that there is too much variability to be able to give a standard name to each possibility that distinguishes it from all others. The number of faces alone is not enough.

I would describe it as removing 5 faces from each of 12 icosahedra, and joining them to the edges of a dodecahedron. You might call it a dodeca-pentadecahedron, meaning 12 sets of 15 faces. There may actually be an existing name, perhaps here, but I don't have time to figure it out.