glossary

Higher-dimensional geometry (previously "Polyshapes").

glossary

Postby alkaline » Sun Nov 23, 2003 5:48 pm

i've revamped the glossary. Here are the changes i made (this list is also on the front page):

- reworded almost all of the definitions
- removed phi/beta, lambda/rho, hemiglome, tetracube
- added "obvious" terms to make glossary more complete, and make it easier to find fourth dimension terms if you know the 2d or 3d terms.
- added bionian, trionian, tetronian; apos, zakos; column, row, pillar; hyperplane, omniverse, multiverse, universe
- added cardinal directions

any comments/complaints/suggestions for the new glossary?
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n-dimensional shapes

Postby Aale de Winkel » Tue Nov 25, 2003 6:16 pm

The glossary mentions a lot of objects, with this posting I'll attempt to put those objects with mathematical formulae.
readers using other terms then noted might send me a personal mail to correct / add to the below listed objects.
The used coordinate (x,y,z,w) are relative to the orientation of the object;
(the reader must see this as a nondefinite discussion of these objects, your comments are appreciated)
notation d just stands for a distance (might be corected to δ in some future upload)

hyperellipse: (x[sub]k[/sub] a[sub]k[/sub] : (x/a)[sub]k[/sub] (x/a)[sup]k[/sup] = r[sup]2[/sup])
2d: ellipse : x[sup]2[/sup] / a[sup]2[/sup] + y[sup]2[/sup] / b[sup]2[/sup] = r[sup]2[/sup]
3d: elipsoid: x[sup]2[/sup] / a[sup]2[/sup] + y[sup]2[/sup] / b[sup]2[/sup] + z[sup]2[/sup] / c[sup]2[/sup] = r[sup]2[/sup]
4d: tesserellipse : x[sup]2[/sup] / a[sup]2[/sup] + y[sup]2[/sup] / b[sup]2[/sup] + z[sup]2[/sup] / c[sup]2[/sup] + w[sup]2[/sup] / d[sup]2[/sup] = r[sup]2[/sup]
(note:the einstein summation in the general formula needs some interpretation, the explicit clarifies)

hypersphere: (x[sub]k[/sub]: x[sub]k[/sub] x[sup]k[/sup] = r[sup]2[/sup])
2d: circle : x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup]
3d: sphere: x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] = r[sup]2[/sup]
4d: glome : x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] + w[sup]2[/sup] = r[sup]2[/sup]

hyperball: (x[sub]k[/sub]: x[sub]k[/sub] x[sup]k[/sup] <= r[sup]2[/sup])
2d: disc: x[sup]2[/sup] + y[sup]2[/sup] <= r[sup]2[/sup]
3d: ball: x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] <= r[sup]2[/sup]
4d: gongyl: x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] + w[sup]2[/sup] <= r[sup]2[/sup]

(the vectors below are monagonal vectors, so a single +/-1 in those (not conform my notation))
hyperbeam: rectangular shaped figure (all faces are rectangle) ([[sub]j[/sub]0]<[sub]k[/sub]1>;[[sub]j[/sub]d[sub]j[/sub]]<[sub]k[/sub]-1>)
2d: rectangle : 1 face ([0,0]<1,0>;[0,0]<0,1>;[dx,dy]<-1,0>;[dx,dy]<0,-1>)
3d: beam : 6 faces ([0,0,0]<1,0,0>;[0,0,0]<0,1,0>;[0,0,0]<0,0,1>;[dx,dy,dz]<-1,0,0>;[dx,dy,dz]<0,-1,0>;[dx,dy,dz]<0,0,-1>)
4d: tesserbeam: 12 faces (simular with w-direction added)

hypercube: rectangular equal length shaped figure (all faces are squares) ([[sub]j[/sub]0]<[sub]k[/sub]1>;[[sub]j[/sub]d]<[sub]k[/sub]-1>)
2d: square : 1 face ([0,0]<1,0>;[0,0]<0,1>;[d,d]<-1,0>;[d,d]<0,-1>)
3d: cube : 6 faces ([0,0,0]<1,0,0>;[0,0,0]<0,1,0>;[0,0,0]<0,0,1>;[d,d,d]<-1,0,0>;[d,d,d]<0,-1,0>;[d,d,d]<0,0,-1>)
4d: tessercube: 12 faces (simular with w-direction added)

{NOTE: this piece if edited after the quotation by bobxp:}
the qualitative descriptions of the 4d hypercylinders are a bit foggy:
hypercylinder: (x[sub]k[/sub]: x[sub]k[/sub] x[sup]k[/sup] = r[sup]2[/sup]; k = 1..l; linear for k > l; l < n)
3d: cylinder: x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup], z = 0 .. dz
4d: cubinder: x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup], z = 0 .. dz, w = 0 .. dw
4d: spherinder: x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] = r[sup]2[/sup], w = 0 .. dw
(note last edit changed the range of k (for 4d either 2 or 3) the following is outside the general formula but is currently classified as a hypercylinder by Garrett)
4d: duocylinder: x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup], z[sup]2[/sup] + w[sup]2[/sup] = r[sup]2[/sup]
(note: I do think that I interpreted the description of cubinder and spherinder correctly, the duocylinder was corrected by Garrett)

hypercone: (x[sub]k[/sub]: x[sub]k[/sub] x[sup]k[/sup] = x[sub]l[/sub][sup]2[/sup]; k = 1,l-1; linear for k >= l)
3d cone: x[sup]2[/sup] + y[sup]2[/sup] = z[sup]2[/sup], z = 0 .. dz
4d cubone: x[sup]2[/sup] + y[sup]2[/sup] = z[sup]2[/sup], z = 0 .. dz, w = 0 .. dw
4d sphone: x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] = w[sup]2[/sup], w = 0 .. dw
(note this is a guess here based on the glossaries qualitative description)
NEW: the general hypercone fumula listed alowed for the shape I called cubone, simular to the cubinder!!

hyperplane: (Garrett noted me of mishaps here earlier, I incorperated those in some editing)
(((NOTE: the equations now listed are the equations for the hyperplane in the space of 1 higher dimension)))
1d: line: [x,y] + λ <dx,dy>
2d: plane: [x,y,z] + λ <dx1,dy1,dz1> + μ <dx2,dy2,dz2>
3d: realm: [x,y,z,w] + λ <dx1,dy1,dz1,dw1> + μ <dx2,dy2,dz2,dw2> + ν <dx3,dy3,dz3,dw3>
4d: flune: equation in pentaspace not postulated!
(note: vectors need be lineairly independent)

monagonal directions in a hyperbeam (amount of trailing 0's depends on dimension)
row: <[sub]1[/sub]1> ie <1,0....0>
column: <[sub]2[/sub]1> ie <0,1,0...0>
pillar: <[sub]3[/sub]1> ie <0,0,1,0...0>

going through the glossary yet to be formulated:
crind

NOTE: this is a first attempt at formulating more precicely the shapes mentioned in the glossary.
writing it I noticed that more combinations might be formulated. Aside from the hyperellips other
conic sections might be utilised, rotating a hyperbol x[sup]2[/sup] / a[sup]2[/sup] - y[sup]2[/sup] / b[sup]2[/sup] = r[sup]2[/sup]
around the z / w axis etc. are possible.
(all these stuff I'll leave for future scruteny, the reader is invited to contribute)

edits: except for alterations to hypercylinder and hyperplane edits where merely cosmetic. see for hypercylinder reasoning postings below.
Last edited by Aale de Winkel on Wed Nov 26, 2003 4:31 pm, edited 11 times in total.
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Re: n-dimensional shapes

Postby Keiji » Tue Nov 25, 2003 10:35 pm

Aale de Winkel wrote:the qualitative descriptions of the 4d hypercylinders are a bit foggy:
hypercylinder: (x[sub]k[/sub]: x[sub]k[/sub] x[sup]k[/sup] = r[sup]2[/sup]; k = 1,2; other for k > 2)
2d: same as circle
3d cylinder: x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup], z = 0 .. dz
4d cubinder: x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup], z = 0 .. dz, w = 0 .. dw
4d duocylinder: x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup], z[sup]2[/sup] + w[sup]2[/sup] = r[sup]2[/sup]
4d spherinder: x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] = r[sup]2[/sup], w = 0 .. dw


A 4-D hypercylinder is a cubinder.
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hypercylinders

Postby alkaline » Wed Nov 26, 2003 2:22 am

well, it depends on how you define hypercylinder. I define it as a rotatope that isn't a hypercube or a hypersphere. That leaves everything in between to be a hypercylinder. They are basically shapes that aren't either flat on every side or perfectly round within its dimensional space.
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formulae

Postby Aale de Winkel » Wed Nov 26, 2003 6:52 am

whether or not duocylinder and spherinder are hypercylinders is indeed a matter of how one defines the hypercylinder (is there perhaps a practising mathematician amongst us(?)), from the formula of the duocylinder it is unclear how it combines into one shape, I assume there is some connection. The spherinder and cubinder forms two seperate generalisations of the regular cylinder, and for me are equaly valid.
according to the listed general formula (x[sub]k[/sub]: x[sub]k[/sub] x[sup]k[/sup] = r[sup]2[/sup]; k = 1,2; other for k > 2) only the cubinder satisfy this, so the limit on k might well be k = 1..l, which allows for the spherinder. (in this case the duocylinder is out of the hypercylinder series, and must be moved to some other class of objects)
I edited the text accordingly, (I do now think the duocylinder must be moved to a separate class of objects)
also removed the remark "2d same as circle" because just as the "duocylinder" it has no linear component, as the now listed general formula states the definition of the hypercylinder is linair components attaches to circular objects, and so forks with 4d into the cubinder and spherinder.
another object as this exist: x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup]; y[sup]2[/sup] + z[sup]2[/sup] = r[sup]2[/sup]; w = 0 .. dw (Garrett suggested this to be the "crind");
that 3d figure (pe w=0) however seems to be 2 circles on the planes (x,y,+/-x), so is a combination of two 2 dimensional figures, so most likely NOT the intended "crind"!
Anyhow these kind of hyperspheroids(?) are also outside the hypersphere and hypercylinder general formulae! So perhaps the duocylinder starts this new type, since the above formulated seems to nillify itself.

Take note that the formulae are still preliminairy, also other possibilities exist, how about for instance:
x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] - w[sup]2[/sup] = r[sup]2[/sup]
x[sup]2[/sup] + y[sup]2[/sup] - z[sup]2[/sup] - w[sup]2[/sup] = r[sup]2[/sup]
x[sup]2[/sup] + y[sup]2[/sup] - z[sup]2[/sup] = w[sup]2[/sup]
I haven't the faintest what these represent, it combines the sphere with the parabole, also combinations of ellipses with hyperboles are possible.
feel free to figure these kind of shapes out, I currently haven't time to do so. It might well be that these amount to nothing, but who knows!
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Postby Keiji » Wed Nov 26, 2003 4:36 pm

A hypercylinder is a n-d solid with one hyperspherical face.
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Postby alkaline » Wed Nov 26, 2003 4:45 pm

it's kind of hard to define "face" for rotatopes; for example, if you look at the cylinder, the flat end surfaces are easily faces, but then you would have to call the part that wraps around the cylinder a face also. The end faces are circles but the wrap around part is a rectangle. Are you saying that you could define a hypercylinder as having at least one "set" of flat hyperspherical faces?
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Postby Keiji » Wed Nov 26, 2003 5:28 pm

alkaline wrote:the wrap around part is a rectangle


Not neccaserily, it could be a parallelogram

Oh and yes I did mean set and only *ONE* of them
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Postby Aale de Winkel » Wed Nov 26, 2003 6:25 pm

bobxp wrote:A hypercylinder is a n-d solid with one hyperspherical face.


To me it's just how one generalises the 3d-equations
the cubinder extents the linear part
the spherinder extents the circular part

with the introduction of the cubone also the hypercone's have 2 4d member, aside from the 3d member which we regularly know.

the cub- extention has in 4d 1 circular face
the sph- extension in 4d has a sperical portion

both might in a way be called "hyperspherical face" so no problem here!
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-ension

Postby alkaline » Wed Nov 26, 2003 6:49 pm

how about the series:

3d:
square base = pyramid

4d:
pyramid extension = ?
cube base = cubamid?

maybe we could thinking of a systematic ending for extending something into a higher dimension - maybe "ension". So:

tesseract = cubension
cubinder = cylension
spherinder = spherension
cubone = conension
pyramid extension = pyrension

and maybe we don't have to replace the already cemented words (like tesseract, cubinder, spherinder) but we could use it for new shape names.
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Postby Keiji » Wed Nov 26, 2003 9:26 pm

cubone


?????

And I have made a term for a 4 dimensional pyramid: the hyperpyramid (ie, a sphere-based hyperpyramid for example)

You can even base hyperpyramids off of pyramids, eg:

a "square based pyramidal hyperpyramid" - a "square" is conically extruded to form a "square based pyramid", which is then conically extruded to form a "square based pyramidal hyperpyramid".

You can also do this with other things, eg:

a "hexagonal prismidal hyperpyramid" - a "hexagon" is linearly extruded to form a "hexagonal prism", which is then conically extruded to form a "hexagonal prismidal hyperpyramid".

I just realised I've made quite a few shape-developing terms:

lathing
conical extrusion
linear extrusion
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Re: n-dimensional shapes

Postby Aale de Winkel » Thu Nov 27, 2003 8:48 am

Guys in these kind of namecalling I get lost, I only extend items by formulae. Changing "hyper-" into "-ension" is a bad idea.
Lots of things are possible, see the polyhedron-dude site(?) (currently must do that myself still (I couldn't read it at home, and couldn't access it at work))

whether "cubone" is the right term I don't know but as a "cub(ic extension to a c)one" it made sense to me. If the thing has another name already please let me know (I'm not familiar with existing literature on these matter)
Note that: cubinder "cub(ical extension to a cyl)inder", spherinder "spher(ical extension to a cyl)inder" and sphone "sph(erical extension to a c)one" are derived simular! :shock:

a "con(ic extension to a c)one" or conone I haven't formulated yet(?).
this conone would make a third 4d member to the hypercones, as would the "con(ic extension to a cyl)inder" (or coninder) would make a third hypercylinder, if one allows the "linear for k > l" to move into a single point!
as to the conone I see among others the following two possibilities:
x[sup]2[/sup] + y[sup]2[/sup] = z[sup]2[/sup], x[sup]2[/sup] + w[sup]2[/sup] = y[sup]2[/sup] and
x[sup]2[/sup] + y[sup]2[/sup] = z[sup]2[/sup], x[sup]2[/sup] + w[sup]2[/sup] = z[sup]2[/sup]
so the conone is not definitely defined by the contraction of the name "con(ic extension to a c)one", how this extension is done is certainly relevant. Simular several possibilities for the coninder.
probably though: x[sup]2[/sup] + y[sup]2[/sup] = z[sup]2[/sup], x[sup]2[/sup] + z[sup]2[/sup] = w[sup]2[/sup] makes the figure crunch to a point at w = 0 so makes the true "conone"?
and: x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup], x[sup]2[/sup] + z[sup]2[/sup] = w[sup]2[/sup] the true "coninder"???(though has points (0,+/-r,0,0)????) :lol:

Aale de Winkel wrote:hypercylinder: (x[sub]k[/sub]: x[sub]k[/sub] x[sup]k[/sup] = r[sup]2[/sup]; k = 1..l; linear for k > l; l < n)

hypercone: (x[sub]k[/sub]: x[sub]k[/sub] x[sup]k[/sup] = x[sub]l[/sub][sup]2[/sup]; k = 1..l-1; linear for k >= l; l <= n)



the above 2 quoted formulae give 1 3d object and 2 4d objects, which make a lot of sense to me.

The "duocylinder" is quite an interesting object, however is not a cylinder, ever point on an (x,y) circle is on an (z,w) circle and vice versa. An object that can exist in >= 4d - spaces. :idea:

Note that I only wanted to make sense of the objects already listed in the glossary.
The qualitative description of crind amounts to an intersecting:
{ x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup] ; x[sup]2[/sup] + z[sup]2[/sup] = r[sup]2[/sup] }
or something simular. :?:
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tetra conicals

Postby Aale de Winkel » Thu Nov 27, 2003 5:23 pm

the duocylinder point can be written in polar coordinates:
x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup], z[sup]2[/sup] + w[sup]2[/sup] = r[sup]2[/sup] ==> r [ sin(α), cos(α), sin(β), cos(β) ]

the following equations form conical shapes in w:
x[sup]2[/sup] + y[sup]2[/sup] = w[sup]2[/sup], x[sup]2[/sup] + z[sup]2[/sup] = w[sup]2[/sup] ==> w [ sin(α), cos(α), cos(α), 1 ]
x[sup]2[/sup] + y[sup]2[/sup] = w[sup]2[/sup], x[sup]2[/sup] + z[sup]2[/sup] = y[sup]2[/sup] ==> w [ sin(α), cos(α), cos(α) sqrt(1-tan[sup]2[/sup](α), 1 ]
x[sup]2[/sup] + y[sup]2[/sup] = w[sup]2[/sup], z[sup]2[/sup] + w[sup]2[/sup] = y[sup]2[/sup] ==> w [ sin(α), cos(α), sqrt(cos[sup]2[/sup](α)-1), 1 ]

all these are single parameter 3d shapes which inflates along the w-coordinate ?.
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Postby alkaline » Fri Nov 28, 2003 2:57 pm

bobxp wrote:And I have made a term for a 4 dimensional pyramid: the hyperpyramid (ie, a sphere-based hyperpyramid for example)

I was talking specifically about the square based one. The problem with the hyper prefix is that depending on its context, it usually means either 4+ dimensions or n dimensions, and it doesn't mean only =4 dimensions very often.

bobxp wrote:You can even base hyperpyramids off of pyramids, eg:

a "square based pyramidal hyperpyramid" - a "square" is conically extruded to form a "square based pyramid", which is then conically extruded to form a "square based pyramidal hyperpyramid".

You can also do this with other things, eg:

a "hexagonal prismidal hyperpyramid" - a "hexagon" is linearly extruded to form a "hexagonal prism", which is then conically extruded to form a "hexagonal prismidal hyperpyramid".

These giant compounds become unwieldy very quickly. That's why we've used the practice of "blend" compounds.

bobxp wrote:I just realised I've made quite a few shape-developing terms:

lathing
conical extrusion
linear extrusion


here are some fun word suggestions:
contrude = con(icly ex)trude
lintrude = lin(early ex)trude
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Postby Keiji » Fri Nov 28, 2003 5:34 pm

alkaline wrote:
bobxp wrote:And I have made a term for a 4 dimensional pyramid: the hyperpyramid (ie, a sphere-based hyperpyramid for example)

I was talking specifically about the square based one. The problem with the hyper prefix is that depending on its context, it usually means either 4+ dimensions or n dimensions, and it doesn't mean only =4 dimensions very often.


Well you can guarantee that whenever I say the prefix hyper, I'm referring to a 4d object.

bobxp wrote:You can even base hyperpyramids off of pyramids, eg:

a "square based pyramidal hyperpyramid" - a "square" is conically extruded to form a "square based pyramid", which is then conically extruded to form a "square based pyramidal hyperpyramid".

You can also do this with other things, eg:

a "hexagonal prismidal hyperpyramid" - a "hexagon" is linearly extruded to form a "hexagonal prism", which is then conically extruded to form a "hexagonal prismidal hyperpyramid".

These giant compounds become unwieldy very quickly. That's why we've used the practice of "blend" compounds.


What's wrong with them? Sometimes you have to be specific... :wink:

bobxp wrote:I just realised I've made quite a few shape-developing terms:

lathing
conical extrusion
linear extrusion


here are some fun word suggestions:
contrude = con(icly ex)trude
lintrude = lin(early ex)trude


Nice.
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Postby Polyhedron Dude » Mon Dec 08, 2003 6:40 am

bobxp wrote:
cubone

And I have made a term for a 4 dimensional pyramid: the hyperpyramid (ie, a sphere-based hyperpyramid for example)

You can even base hyperpyramids off of pyramids, eg:

a "square based pyramidal hyperpyramid" - a "square" is conically extruded to form a "square based pyramid", which is then conically extruded to form a "square based pyramidal hyperpyramid".

You can also do this with other things, eg:

a "hexagonal prismidal hyperpyramid" - a "hexagon" is linearly extruded to form a "hexagonal prism", which is then conically extruded to form a "hexagonal prismidal hyperpyramid".


The hyper prefix actually isn't needed, for example - a cube based pyramid (or cube pyramid) is obviously a 4-D object. A pyramid of a pyramid could be called a dipyramid or better yet a "diamid". A good name for the hexagonal prismidal hyperpyramid would be hexagon prismid - I've actually used the word "prismid" for this sort of thing. Another interesting case is the "duopyramids" - or better yet "duomids" - an example is the pentagon-hexagon duomid - put the pentagon on the xy plane and the hexagon on the zw plane, then separate the two planes into a 5th dimension - its as though one polygon is the apex while the other is the base - or vise versa. A P diamid is actually a P-edge duomid (P representing any shape). We could also have trimids, tetramids, pentamids, etc.

Picture this object - the cube-octagon-tesseract-gaquatid tetramid - the cube is in dimensions 1-3, the octagon in dimensions 4 and 5, the tesseract 6-9, and gaquatid (a starry uniform polyhedron) in dimensions 10-12. The cube pans outwards into dimension 13, the octagon then goes out into dimension 14, the tesseract then separates from gaquatid into dimension 15 - producing some crazy 15-D polytope.

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Postby Keiji » Mon Dec 08, 2003 5:08 pm

oh.... my......................... god...........................................................

that is confusing :|
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Postby sup2069 » Tue Dec 09, 2003 3:55 pm

bobxp wrote:oh.... my......................... god...........................................................

that is confusing :|



Dont feel bad, I got lost on the end of the second sentence.
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Postby Keiji » Fri Dec 12, 2003 10:27 pm

:buMp:

Please add these:

extrude (to form a prism of)
contrude (to form a pyramid of)
contract (to take the most significant n-face of, opposite of extrude; e.g. contracting a hexagonal prism would give a hexagon; contracting a dodecahedron would give a pentagon)
lathe (to take the object's sweep area)

and fix the table under tetraspace (it says under tetronian to see the table under tetraspace, but the table doesn't mension anything about tetronians)
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Postby alkaline » Fri Dec 12, 2003 10:29 pm

ok, those are due for the next glossary update.

what do you think about "pointrude" instead of "contrude"? The word "pointrude" just seems to vividly call up the image in my head.
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Postby Keiji » Fri Dec 12, 2003 10:31 pm

no - i don't like that, i prefer contrude, it sounds much better :|
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Postby Aale de Winkel » Thu Jan 08, 2004 10:36 am

In case you post definite words for certain kinds of extrusions onto the glossary do define them exactly, I tried to do this on my own page on the matter:
http://home.wanadoo.nl/aaledewinkel/Enc ... hapes.html
which defines the "extrusion", as the perpendicular variety (to create polygon-rods), and the "anti-extrusion" (which creates polygon-anti-rods) (Generic terms I defined on that page for the objects Polyhedron Dudes defined)
"coning" also has two distinctive varieties "symmetric" and "non-symmetric" depending whether or not the "cone-point" is perpendicular to the coned objects center. These two varieties definitely create different objects.

(strange first tripple digit posting, but here it is :lol: )
Aale de Winkel
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