4-Dimensional Objects

[Elephant]

my question is how could you make the d20 system work without a d20?

This part is easy: Multiply everything by 5 and use percentile rolls. It has the added advantage of reducing crits and fumbles to 1% rather than 5% of the time.

 

[arscott]

This part is easy: Multiply everything by 5 and use percentile rolls. It has the added advantage of reducing crits and fumbles to 1% rather than 5% of the time.

How is that an advantage? I likes my critical hits. (and don't use fumbles, for that matter.)

 

[Li Shenron]

I do believe he's referring to n-sided 3d objects being extruded into 4d (or possibly higher).

I think some of the difficulty arises in the method of "Extrusion" used. Consider a square, and a circle. Brought from 2d to 3d, the square becomes a cube, and generally you think that the cirlce becomes a sphere. Except that if you use the same extrusion method on the circle as the square, you'll get a cylinder instead. If you use the same extrusion method on the square as you used to get a circle into a sphere, you'll... also get a cylinder. In order to produce both a cube and a sphere, you need to use different methods.

So... what method are you using the extrude your hyperpyramid? Hyperdiamond? Hyperdecahedron? Hyperduodecahedron? Etc? The method used will alter the topography drastically, as seen in my 2d->3d example above. Unless you can define the translation, you can't define the number of sides.

I am not a matematician, but by instinct I wouldn't think of extrusion methods based on rotation to be a good choice here.

What comes to my mind is something related to defining the 3D shapes from a property and then apply the same definition in 4D and try to "see" what comes out of it. For example, a sphere in 3D could be defined as "the space inside a surface whose points all have distance X from a special point in space called centre". This is also the same property which defines a circle (the 2D equivalent of a sphere). So you can probably call "hypersphere" the 4D figure which has this same property.

The work to do could be do write down what is the defining property of different shapes, and this for me is not a feasible task. But I think the number of sides isn't a defining property, and besides I am not at all sure that there IS any hyper-something with as few as 6 sides! You know... 3 is minimum sides in 2D, and 4 is minimum sides in 3D, who knows what's the minimum sides in 4D? I bet that at least you just cannot have a d4 in 4D

 

[Steverooo]

n-dimensional cube has (n!/k!(n-k)!)*2^(n-k) k-dimensional components (cubes).
ie. 3-dim d6 has (3!/2!(3-2)!)*2^(3-2) = (6/2)*2 = 6 2-dim (square) sides, (3!/1!(3-1)!)*2^(3-1) = (6/2)*4 = 12 1-dim (line) edges and (3!/0!(3-0)!)*2^(3-0) = (6/6)*8 = 8 0-dim (point) vortices.

Something is whack, here... If this formula is correct, as typed, then N-K should be computed, first, then the 2 exponentiated, then N!/K! and then MULTIPLIED by (N-K)!, shouldn't it? Should it, perhaps, be: (N!/(K!(N-K)!)*2^(N-K)? (That seems to be the way that you're applying it...)

In either case, I am not getting the correct numbers for the Hypercube...

Anyway, for 3D solids, S = E - V + 2. A cube has six sides, 12 edges, and 8 vertices. 6=12-8+2. For the triangular-based pyramid (D4) it's 4=6-4+2. For the square-based pyramid it's 5=8-5+2. It would be interesting to derive the formula for polychorons... if I could get this formula to kick out the right numbers!

So zero factorial (0!) is one? (Been too long since I used'em!) Am I forgetting something about how to do factorials that's messing me up on the formula?

I iz flustrated! (and no, that is not a typo!)

 

[Elephant]

Arscott, it seems stupid to some of us that the most experienced, accomplished warrior in existence, sparring with a series of farmers' children who barely know how to hold a sword, will fail to strike them a full 5% of the time. The d20 math doesn't mesh with what feels right.

 

[Goblyn]

So zero factorial (0!) is one? (Been too long since I used'em!) Am I forgetting something about how to do factorials that's messing me up on the formula?

I think it is understood as such, but it's been a while since I've touched the stuff.

I iz flustrated! (and no, that is not a typo!)

You're right; it's two typos.

 

[Cor Azer]

So zero factorial (0!) is one? (Been too long since I used'em!) Am I forgetting something about how to do factorials that's messing me up on the formula?

Yay! I can contribute something.

Yes, 0! is arbitrarily set to 1.

Li Shenron refered to the properties of various regular objects in N-dimensions. I think what you're refering to are the algebraic formulae for these shapes. Ie. A circle in 2D is defined by a^2 + b^2 = c^2 (in simplest form). Similarly, a sphere is a^2 + b^2 + c^2 = d^2. By extension, a hypershere would be a^2 + b^2 + c^2 + d^2 = e^2.

 

[der_kluge]

unavailable for comment:

*


the best representation of 4 dimensions I can think of. vertical, horizontal, forward/backward, and time (scrolling).

 

[Torm]

the best representation of 4 dimensions I can think of. vertical, horizontal, forward/backward, and time (scrolling).

Right. I'm not a stupid person, or at least I don't think so, when it comes to these things, and yet I have been left by most of this thread puzzled as to why anyone would think that a "4D" object would have any more sides than a "3D" object when the 4th dimension is an expression of transition through TIME rather than spacial dimensionality......?

 

[WayneLigon]

...why anyone would think that a "4D" object would have any more sides than a "3D" object when the 4th dimension is an expression of transition through TIME rather than spacial dimensionality......?

If I remember right - and I'm not sure I've kept up to current thought - it's because time is not the fourth dimension; time and space are one indivisible thing, so extra dimensions are all both spacial and temporal dimensions.

Ahh, 's gone all quantum and miffic now....