Wouldn't continually cutting off the corners result in more faces, not fewer? Eventually you'd end up with something that would, to the naked eye, appear to be a sphere, like a d100.'First it's a cube' (chopping) 'and now we cut off the corners' (more chopping) 'and it's an octahedron. And now if we cut off the corners again' (more chopping) 'it's back to a cube. Okay everyone, mashed potato for dinner tonight!'
Depends. Are you cutting it for Spud Wars (and the old version needed a d6 so that was easy) or for Vilyams and Vigilantifries?
Six slices can turn ANY solid into a cube. What I don't understand is how this is an inherent property of octahedrons.
Director of Brand Strategy
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Six slices can turn ANY solid into a cube. What I don't understand is how this is an inherent property of octahedrons.
If you are specifically making slices in order to cut off corners, an octahedron will require more than six cuts. making only six slices and resulting in a cube would require that at least some of the cuts take off more than one corner at a time.
Six slices can turn ANY solid into a cube. What I don't understand is how this is an inherent property of octahedrons.
An octahedron has 6 corners (vertices) so it's one cut per corner.
a cube has 8 corners, so 1 cut per corner transforms it into an octahedronEdit: ninja'd by yoda, that litte green goblin
An octahedron has 6 corners (vertices) so it's one cut per corner.
Ah, okay, I get it now.
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Director of Brand Strategy
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Six slices can turn ANY solid into a cube. What I don't understand is how this is an inherent property of octahedrons.An octahedron has 6 corners (vertices) so it's one cut per corner.
a cube has 8 corners, so 1 cut per corner transforms it into an octahedronEdit: ninja'd by yoda, that litte green goblin
I was actually incorrect in my statement above. I guess it all depends on whether you are making identical faces with subsequent cuts in terms of what type of polyhedron it makes. It's possible to cut the corners off an octahedron and end up with 14 sides, though they would be different sizes and shapes. Similarly, you can cut the corners off a cube and end up with the same number, though they will also not be equal faces.
I was actually incorrect in my statement above. I guess it all depends on whether you are making identical faces with subsequent cuts in terms of what type of polyhedron it makes. It's possible to cut the corners off an octahedron and end up with 14 sides, though they would be different sizes and shapes. Similarly, you can cut the corners off a cube and end up with the same number, though they will also not be equal faces.Six slices can turn ANY solid into a cube. What I don't understand is how this is an inherent property of octahedrons.An octahedron has 6 corners (vertices) so it's one cut per corner.
a cube has 8 corners, so 1 cut per corner transforms it into an octahedronEdit: ninja'd by yoda, that litte green goblin
Ah like pie. A lot.
EDIT: And cookies.
Counting down till 10pm tomorrow!
Yes thats right I have to wait till 10 unholy PM to find out I have not made the top 32. Cant wait.
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Counting down till 10pm tomorrow!
Yes thats right I have to wait till 10 unholy PM to find out I have not made the top 32. Cant wait.
Yes, well, at least you'll be at home then. For me, it'll be 4:00 PM and I'll be half-way through my commute home.
Although if you make first round, I could see you having some trouble getting to sleep...
Counting down till 10pm tomorrow!
Yes thats right I have to wait till 10 unholy PM to find out I have not made the top 32. Cant wait.
Yes, well, at least you'll be at home then. For me, it'll be 4:00 PM and I'll be half-way through my commute home.
Although if you make first round, I could see you having some trouble getting to sleep...
Indeed. By then my children are in bed and I am having some net time. I may have to brainstorm before even attempting to sleep. I may have to do that anyway to see what I come up with even if a dont make it (just to keep up with everyone else)
Ah like pie. A lot.
EDIT: And cookies.
Ah like turtles. *zombie stares at the camera*
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Director of Brand Strategy
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Counting down till 10pm tomorrow!
Yes thats right I have to wait till 10 unholy PM to find out I have not made the top 32. Cant wait.
It's the same number of hours everyone has to wait, so your local time doesn't mean you find out later. You're already halfway through Monday afternoon, while they haven't even woken up in Seattle yet. But everyone has about 30 hours to go.
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+1. For the record, I'd love to be one of the top 32 and get to continue, but I already feel like I've "won" because I introduced my item into my current campaign and my players are enjoying finding all kinds of uses for it.Along this line, there's at least one Top 32 submission that made us say, "we need to include a suite of items like this in the Advanced Player's Guide!" :)
I'm only reading this right now. It just makes my day and lets me dream for 24+ odd hours it could be mine.
Yes - there could be a whole suite of my 'item' - from weak and non-magical to what I submitted as a basic low powered wondrous item towards some really elaborate and much more powerful ones. Even shape and origin could be altered to form a suite.
Just dreaming - odds are the remark above has nothing at all to do with my submission.
Anyhow - I guess half the people submitting items are convinced their item is missing in the rule-book. After all - you have to believe in what you have done.
Good luck to everyone and especially the author of the wondrous item mentioned by Sean.
Thod
It's the same number of hours everyone has to wait, so your local time doesn't mean you find out later. You're already halfway through Monday afternoon, while they haven't even woken up in Seattle yet. But everyone has about 30 hours to go.Counting down till 10pm tomorrow!
Yes thats right I have to wait till 10 unholy PM to find out I have not made the top 32. Cant wait.
that is true, but think of it this way: when I wake up tomorrow at 6am I have to wait until 10pm. when someone in Seattle wakes up tomorrow at 6am they only have to wait until 2pm.
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Six slices can turn ANY solid into a cube. What I don't understand is how this is an inherent property of octahedrons.An octahedron has 6 corners (vertices) so it's one cut per corner.
a cube has 8 corners, so 1 cut per corner transforms it into an octahedronEdit: ninja'd by yoda, that litte green goblin
I seem to be a little bit of slow ...
I assume you start with a sphere
Starting with faces:
You need one cut per face - simple. This is one option. This makes 4 cuts for a tetrahedron, six for a cube and eight for an octahedron.
Now take corners.
For each (non-degenerate) corner you need 3 cuts - or for degenerate ones as many cuts as edges join that corner. On the plus side - one cut could touch multiple corners.
For the tetrahedron you have 4 corners. Each cut along the faces touches 3 corners. This makes one full corner per cut - which again is 4 cuts. For the cube each cut along the faces touches 4 corners. so each cut yields 1 1/3 corner. As I have to form 8 of these - again - the result is 6 total.
The octahedron seems to be doable with 6 cuts. Per the logic above - I do have 6 corners. Each cut along a face touches three of them. But cutting along the faces can't be the solution as the corners are degenerate and need four cuts to form as four edges join at each corner.
So are there alternatives. I still can't see how to form 8 non-parallel faces with <8 cuts. Or do you start from a different shape as a spud. Turning a cube into an octahedron should need less cuts as you can re-use already existing faces.
Still wondering. At least - I plan to be at the Paizo-Con UK - so you could show me.
Thod
Edit - just re-read where it started. Charles Evans only said you can turn a cube into an octahedron by cutting off the corners. The number six slipped in later and I assumed you could get an octahedron using six cuts. Well - as the cube has 8 corners - we are back to 8 cuts. The only helpful bit is - it is easier to cut a symmetric cube out of a spud as an octahedron. So by cutting a symmetric cube first it should ease to get a symmetric octahedron by using the corners of the cube as guidance where to cut. Did I now get it?
Is all of this talk about cutting facets taking the edge off the wait? *wink*
I can't quite follow you, but perhaps it helps you to understand the whole cutting corners business as an analogy for remapping the centers of each face of the object to corners of the new object.
a cube has 6 identical faces (and 8 corners), if you take the centers of these faces as the corners of a new object you get an octahedron (6 corners, 8 sides) and vice versa. Mathematically this means that their Symmetry Groups are the same.
[ You have 24 possibilities to place a d6 in front of you, with 1 face down and one facing you , and you have (the same) 24 possibilities to place a d8 in front of you with 1 face down and 1 corner of the top face pointing to you ]
Is all of this talk about cutting facets taking the edge off the wait? *wink*
well, it's better than cutting yourself.
Is all of this talk about cutting facets taking the edge off the wait? *wink*well, it's better than cutting yourself.
True. Best leave that bit to the judges!
RPG Superstar 2009 Top 8
aka Tarren Dei
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I do not concern myself with anxiety. I already know that I'm not in the top 32. The rest of you may continue your panicing and wasteful tension. I shall enjoy the calm that comes with knowledge of my failure.
Yeah. I felt the same way last year. It was very reassuring. I was absolutely sure I wasn't getting in.
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Trevor,
You are an evil being.