Naming and notations

Discussion of shapes with curves and holes in various dimensions.

Naming and notations

Postby moonlord » Tue May 30, 2006 12:42 pm

To summarize some aspects I've encountered when thinking about the naming, I've came to this conclusion:

CSG should start with a point. You can extrude it and get a segment, or lathe it and get two points. This way, the number of letters in the shape's CSG notation will reflect its dimensionality.

Now about the naming. Some shapes (I suspect, all do) have different "flavours". For example (CSG abbreviations):

a. A circle is LL or, in Cartesian equations, x2 + y2 = r2. A circle is one-dimensional, as it is, in fact, a "curved" segment.
b. A disk is EL or, in Cartesian equations, x2 + y2 <= r2. A disk is, obviously, two-dimensional.

Now, a circle and a disk are two "flavours" of the same thing, as I call them. Another example:

a. A sphere is LLL or, in Cartesian equations, x2 + y2 + z2 = r2. A sphere is two-dimensional.
b. A ball is ELL or, in Cartesian equations, x2 + y2 + z2 <= r2. A ball is three-dimensional.

And another example:

a. EEE is a full cube (|x| <= h, |y| <= h, |z| <= h). It's 3D.
b. A hollow cube, or a box can not be represented with the actual CSG notation. It is something like LEE+LEE+LEE, with the first lathing oriented along x, the second along y and the third along z. Or any permutation of these. This is why I'd use a parameter for lathing and extruding, to show the direction it is performed in. A box becomes LxEyEz+LyExEz+LzExEy. A box is 2D.
c. A wireframe cube is also bogus. It can not be represented even in the extended CSG. So I ask, whether CSG is really useful.

The problem is, different people seem to be reffering (sometimes) to different things, because of the naming problem. Not to introduce new names, I'd just call them the nD-body, i.e. the 2D cube and so on.

RNS has the same problem. The simplest solution I see is, as I've already mentioned, to specify the dimensionality of the considered body.

I'd also note that line is sometimes incorrectly used for segment. One is finite, one is not.

So, to make a list:

1. point (0D)
2. segment (1D) and hollow segment (0D)
3a. disk (2D) and circle (1D)
3b. square (2D) and wireframe square (1D) and marked square (0D)
3. polygon (2D) and wireframe/hollow/perimeter polygon (1D) and in some cases marked polygon (0D)
4a. ball (3D) and sphere (2D)
4b. cube: full (3D), hollow (2D), wireframe (1D), marked (0D)
4. polyhedron: full (3D), hollow (2D), in most cases wireframe (1D) and in some cases marked (0D)
... and the list can go on.

As you can see, it all refers to whether the body considered has a certain-sub-dimensional component. I'd consider only the full ones when using CSG or RNS notations, but I'd put a link on their explanatory pages to a page explaining this herein.

I saw on the duocylinder wiki page that all these components are considered different "flavours" of the same body. I'd only consider them, well, components.

And with this, we put an end to the discussion whether (21) is a 2-torus or a 3-torus. If it's hollow, it's 2D. If it's full, it's 3D.
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Postby PWrong » Tue May 30, 2006 1:39 pm

CSG should start with a point. You can extrude it and get a segment, or lathe it and get two points. This way, the number of letters in the shape's CSG notation will reflect its dimensionality.

I'm still getting the hang of CSG notation. It's harder to understand a notation you didn't help to invent.

Now about the naming. Some shapes (I suspect, all do) have different "flavours". For example (CSG abbreviations):

I've mentioned this elsewhere, that shapes come in different forms, each of which is a combination of cells. But "flavour" is an interesting word for it.

RNS has the same problem. The simplest solution I see is, as I've already mentioned, to specify the dimensionality of the considered body.

RNS gives us the general shape. If you want to be specific, you can say the "kD form of the ((31)211)21". Most forms are composed of separate cells, each of which is another rotope. Knowing the forms and cells of an object is useful, because there's a nice pascal's triangle structure to the number of cells in each form, and it gives you a way to calculate volumes and areas. For instance, the area of the 2D form of a cylinder is the sum of the areas of its two cells.

I don't usually worry about polygons and polyhedra, but I might do up a table of forms and cells next week. I should be studying for exams, so I'll procrastinate by doing this instead.

I saw on the duocylinder wiki page that all these components are considered different "flavours" of the same body. I'd only consider them, well, components.

I just wrote that page today. What do you mean by components?
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Postby moonlord » Tue May 30, 2006 2:39 pm

PWrong wrote:I'm still getting the hang of CSG notation. It's harder to understand a notation you didn't help to invent.


Well, I didn't help Rob inventing it either. I just applied it. Don't know how to explain better than he can, though...

PWrong wrote:But "flavour" is an interesting word for it.


It's a quite used word. Quarks also come in flavours, for example.

PWrong wrote:For instance, the area of the 2D form of a cylinder is the sum of the areas of its two cells.


Doesn't it have three cells? Two disks and a rolled up rectangle?

PWrong wrote:What do you mean by components?


I'll give an example: two disks, a rolled up rectangle and a cylindrical interior are components of a cylinder (full). Two circles are components of the two disks so, by transitivity, they are components of the cylinder. If you want, a cylinder is a component of itself.
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Postby PWrong » Tue May 30, 2006 3:14 pm

Doesn't it have three cells? Two disks and a rolled up rectangle?

The pair of disks counts as a single cell. If we define things this way we get the nice pascal's triangle structure I mentioned.

I'll give an example: two disks, a rolled up rectangle and a cylindrical interior are components of a cylinder (full).

Ok, so a component is similar to a cell.

Maybe we should go back to the notation I introduced somewhere in this thread
We use Sn for a sphere in nD, and Bn for a ball in nD.

Here's the three forms of a cylinder:
1D form: S2*S1 - one cell
2D form: B2*S1, S2*B1 - two cells
3D form: B2*B1 - one cell

Here's the cubinder, so you can see the pascal's triangle develop; note the number of cells in each form.

1D form: one cell
S2*S1*S1 = four circles

2D form: three cells
B2*S1*S1 = four disks
S2*B1*S1 = two rolled up rectangles along x direction
S2*S1*B1 = two rolled up rectangles along y direction

3D form: three cells
B2*B1*S1 = two solid cylinders along x
B2*S1*B1 = two solid cylinders along y
S2*B1*B1 = rolled up rectangle along x, extruded along y.

4D form: one cell
B2*B1*B1 = solid cubinder
Last edited by PWrong on Wed May 31, 2006 2:53 pm, edited 1 time in total.
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Postby Marek14 » Wed May 31, 2006 5:38 am

CSG notation? What is it? Did I miss something?
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Postby wendy » Wed May 31, 2006 11:19 am

The polygloss already makes the following distinctions:

glomohedrix = surface of sphere vs glomohedron = surface + content.

These are freely usable in product names, eg

biglomohedric prism = S3 * S3
glomohedral glomohedric prism = S3 *B3
biglomohedral prism = B3 * B3

It is interesting that there are four different meanings of * implemented in the PG, prism, tegum, comb and pyramid.

A solid cone is then a teelic glomolatral pyramid, ie point ** circle pyramid product. These extend into, eg 5d

biglomolatric pyramid = circle ** circle.

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Postby moonlord » Wed May 31, 2006 12:14 pm

PWrong wrote:Here's the three forms of a cylinder:
1D form: S2*S1 - one cell
2D form: B2*S2, S2*B2 - two cells
3D form: B2*B2 - one cell


Here's where I lose it. I don't have the time to read the thread you provided right now, but I'll try to do it in the evening. Which of the two cells in the 2D form is which component? I ask because I'd expect the two disks to be B2*S1 and the rolled up rectangle to be S2*B1. Again, isn't the 3D form B2*B1? Or do I misunderstand the * product?

As I see it, S1 is two points, S2 is a circle, B1 is a segment and B2 - a disk. I interpreted the * product as simple cartezian product of two perpendicular components.

PWrong wrote:Here's the cubinder, so you can see the pascal's triangle develop; note the number of cells in each form.

1D form: one cell
S2*S1*S1 = four circles

2D form: three cells
B2*S1*S1 = four disks
S2*B1*S1 = two hollow cylinders along x direction
S2*S1*B1 = two hollow cylinders along y direction

3D form: three cells
B2*B1*S1 = two solid cylinders along x
B2*S1*B1 = two solid cylinders along y
S2*B1*B1 = hard to visualise. The squares are solid, but the circles are hollow.

4D form: one cell
B2*B2*B1 = solid cubinder


Again, I'd expect the 4D form to be B2*B1*B1 (disk extruded two times). How about S2*B1*B1? Isn't it a hollow cylinder, extruded?

I definitely need to read that thread...

wendy: I think the Polygloss would be way easier to read if you didn't use that "th" character...
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Postby PWrong » Wed May 31, 2006 2:50 pm

Here's where I lose it. I don't have the time to read the thread you provided right now, but I'll try to do it in the evening. Which of the two cells in the 2D form is which component? I ask because I'd expect the two disks to be B2*S1 and the rolled up rectangle to be S2*B1. Again, isn't the 3D form B2*B1? Or do I misunderstand the * product?

You're right. I'll edit it now. It should be:

1D form: S2*S1 - one cell
2D form: B2*S1, S2*B1 - two cells
3D form: B2*B1 - one cell

As I see it, S1 is two points, S2 is a circle, B1 is a segment and B2 - a disk. I interpreted the * product as simple cartezian product of two perpendicular components.

That's right.

Again, I'd expect the 4D form to be B2*B1*B1 (disk extruded two times). How about S2*B1*B1? Isn't it a hollow cylinder, extruded?

Yep, I made another mistake there :oops:. And when I said hollow cylinder, I should have said the curled up rectangle.
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Postby moonlord » Wed May 31, 2006 4:08 pm

Well, I don't call it a hollow cylinder because, by "hollow cylinder", I understand all the 2D cylinder.
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Re: Naming and notations

Postby Keiji » Thu Jun 01, 2006 3:32 pm

Sorry I've been off the forum for so long... I've been addicted to StepMania :D

moonlord wrote:CSG should start with a point. You can extrude it and get a segment, or lathe it and get two points. This way, the number of letters in the shape's CSG notation will reflect its dimensionality.


Right.

Now about the naming. Some shapes (I suspect, all do) have different "flavours". For example (CSG abbreviations):

aa. A circle is LL or, in Cartesian equations, x2 + y2 = r2. A circle is one-dimensional, as it is, in fact, a "curved" segment.
b. A disk is EL or, in Cartesian equations, x2 + y2 <= r2. A disk is, obviously, two-dimensional.


That makes sense.

a. EEE is a full cube (|x| <= h, |y| <= h, |z| <= h). It's 3D.
b. A hollow cube, or a box can not be represented with the actual CSG notation. It is something like LEE+LEE+LEE, with the first lathing oriented along x, the second along y and the third along z. Or any permutation of these. This is why I'd use a parameter for lathing and extruding, to show the direction it is performed in. A box becomes LxEyEz+LyExEz+LzExEy. A box is 2D.
c. A wireframe cube is also bogus. It can not be represented even in the extended CSG. So I ask, whether CSG is really useful.


Then we have a new operator H, which does the same as E but only duplicates the object and inserts the hypercells without solidifying it.

Note that both L and H give the 1-sphere.

EL = Solid circle (disk)
HL = Hollow circle

ELL = Solid sphere (ball)
HLL = Hollow sphere

EEL = Solid cylinder
EHL = Hollow cylinder
HHL = Wireframe cylinder (two circles)

EE = Solid square
EH = Hollow square
HH = Marked square

EEE = Solid cube
EEH = Hollow cube
EHH = Wireframe cube
HHH = Marked cube

etc.

I'd also note that line is sometimes incorrectly used for segment. One is finite, one is not.


A line is commonly used both to refer to the infinite line and the finite segment. So I see nothing wrong with this provided it is obvious from the context what you are talking about.

And by the way, to end the argument of nD versus nD, how about:

Bounding space = dimensions required for stating the positions of every point in an object
Net space = dimensions required for referencing every point in an object

So:

Solid square: 2D bounding space, 2D net space
Hollow square: 2D bounding space, 1D net space
Marked square: 2D bounding space, 0D net space

Solid cube: 3D bounding space, 3D net space
Hollow cube: 3D bounding space, 2D net space
Wireframe cube: 3D bounding space, 1D net space
Marked cube: 3D bounding space, 0D net space

etc.

If you approve of this, I will get to changing all the CSG definitions.
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Re: Naming and notations

Postby moonlord » Thu Jun 01, 2006 5:52 pm

Rob wrote:EHL = Hollow cylinder (...)
EH = Hollow square (...)
EEH = Hollow cube
EHH = Wireframe cube


It is a good idea, although it does not actually represent the creation method for the above. That is:

EHL generates the two disks, HLE generates the curled up rectangle, so the hollow cylinder should be EHL+HLE. Same for the others: EH+HE - hollow square, EEH+EHE+HEE - hollow cube, EHH+HEH+HHE - wireframe cube.

The bounding/net space will definitely clear things up. That is, if others agree with the new terminology, of course.

Marek, the wiki will explain it better than I can do.
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Re: Naming and notations

Postby Keiji » Thu Jun 01, 2006 7:09 pm

moonlord wrote:It is a good idea, although it does not actually represent the creation method for the above. That is:

EHL generates the two disks, HLE generates the curled up rectangle, so the hollow cylinder should be EHL+HLE. Same for the others: EH+HE - hollow square, EEH+EHE+HEE - hollow cube, EHH+HEH+HHE - wireframe cube.


Wrong.

E = Line segment
EH = Hollow square
EHL = Hollow cylinder

You have to understand that H doesn't just copy the object, it also inserts the hypercells to connect the two copies together. The difference between H and E is that E also solidifies the resulting object.

Edit: I've changed all the references on the wiki. However, as I cannot think of a way to express the hollow crind, I have used the solid expressions for every shape for consistency.
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Postby moonlord » Fri Jun 02, 2006 11:40 am

Chances are I won't be the only to misunderstand that, so this might need polishing... It also is ambiguous, if this is the right word. Is it the minimum-dimensional cells that are required to link the two copies? Something needs to be "full" (aka, not hollow). What would that be? Why not just copy the component? Why not just use PWrong's notation? Bah, I suspect I'm getting on someone's nerves right now... Don't mind about me...

EDIT: If you do link the copies, then H gives a segment (E) and not two points (L). Also, for example, EEH is not a hollow cube, it is a wireframe cube with two opposite sides full. You do have to define the H transformation in more rigour. The question is, mainly, what kind of cells does it add between the two copies.

EDIT 2: If we take the notation PWrong mentioned and add the operators (reunite, substract, intersect) from CSG, I'd expect to get something good.
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Postby Keiji » Fri Jun 02, 2006 12:42 pm

Assume object x in CSG notation is a solid nD object, e.g. a solid square.

Now the object xH will:

* Duplicate the object and move one copy -1 unit in the (n+1)th dimension and the other copy +1 unit in the (n+1)th dimension.
* Connect together each cell of the first object with the corresponding cell of the other object with an object that is of the same dimension of both bounding and net space as the original object. So if x is hollow 2D, the cells used to connect the objects will also be hollow 2D.

Understand now?
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Postby moonlord » Fri Jun 02, 2006 4:47 pm

From your rather harsh answer I gather I really got on someone's nerves :) ... Anyway, I understand now, but seems too complicated, when you can use more characters with a simpler definition... Just my opinion.
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Postby Keiji » Fri Jun 02, 2006 4:55 pm

It may be complicated, but it simplifies expressions. Also:

EEEE = Solid tesseract
EEEH = Hollow tesseract
EEHH = ? tesseract
EHHH = Wireframe tesseract
HHHH = Marked tesseract

Can anyone think of what the question mark is? :?
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Postby moonlord » Fri Jun 02, 2006 5:17 pm

Well, we do run out of names for higher dimensional objects. How about 0D, 1D, 2D, 3D and 4D 4-cubes :) ?

Or better, let N be the net space dimensionality and B the bounding space dimensionality.

N = 0 - marked
N = 1 - wireframe
N = 2 - ? (What about faceframe?)
N = 3 - ? (What about cellframe?)
N = 4 - ? (What about fluneframe?)
...
N = B-2 - ?
N = B-1 - hollow
N = B - full

I think we should use names for simple (<7D) bodies, and symbols for those in more dimensions. Let N/B-body be the template, if you wish. For example, a wireframe tesseract becomes a 1/4-cube.

Another way would be to call them prefixframe body, where prefix is mono/duo/tri/tetra/penta/hexa/hepta... Or perhaps use prefixes only for N>4. Waiting for ideas and suggestions... :)
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Postby bo198214 » Fri Jun 02, 2006 5:40 pm

fluneframe


I vote for fluffywuffy frame! :lol:
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Postby moonlord » Fri Jun 02, 2006 6:22 pm

Why ironic? That's the only short term I've seen to be used to reffer to a 4D space. Point, line, plane, realm, flune. :?

EDIT: Oh, I just got a visa. Yey...
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Postby Keiji » Fri Jun 02, 2006 6:31 pm

Hmm... I don't like the word marked, so how about:

nullframe
monoframe
biframe
triframe
tetraframe

etc.

So a tetraframe tesseract is a solid tesseract.
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Postby bo198214 » Fri Jun 02, 2006 6:35 pm

moonlord wrote:Why ironic?


Oh it by way not ironic, I only let my child character play! And had a much fun!
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Postby moonlord » Fri Jun 02, 2006 6:38 pm

Rob wrote:nullframe
monoframe
biframe
triframe
tetraframe


Sound nice, however biframe should be duoframe. Sound more "professional" if we don't mix the latin and the greek.
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Postby Keiji » Fri Jun 02, 2006 7:40 pm

Bi means two. It sounds better in this case anyway hehe :P
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Postby houserichichi » Fri Jun 02, 2006 11:40 pm

Rob wrote:nullframe
monoframe
biframe
triframe
tetraframe


Bi and duo are both latin prefixes. The greek equivalents are di or dy. The actual greek numeral 2 is "duo" (or dyo or di) though but the prefix isn't.

Numerical equivalent = Greek prefix -- Latin prefix

1 = mono -- uni
2 = di/dy -- bi/duo
3 = tri -- tri
4 = tetra -- quadri/quart

If we insist on using "tetra" then you must stick with tri, di/dy, and mono for consistency. (think unilateral, bilateral, trilateral, quadrilateral versus monad, dyad, triad, tetrad)
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Postby wendy » Sat Jun 03, 2006 8:42 am

The idiom used in the polygloss for planes and N-space, is to use fabric and patches.

A hedrix is a 2-fabric. One cuts hedrons (2-patches) from it, and make a "polyhedron" (many patches with closure). Likewise, one has 0D teelix, 1D latrix, 3D chorix, 4D terix, 5D petix, 6D ectix, 7D zettix, 8D yottix, to give, eg chorons (3d patches) which give a polychoron.

In the full extend, both hedrix and hedron should take prefixes, which give what fabric or patches actually do. The stems are broken up to show the division of stems. In practice, these are written as one word.

  • glomo-hedrix = globe-shaped 2-fabric (=surface of sphere)
  • horo-hedrix = infinite-radius 2-fabric (flat only in euclidean space)
  • plato-hedrix = straight 2d fabric
  • bollo-hedrix = negative curvature 2-flat

The list of patches are then as follows:

  • sur-hedron = surface 2-d patch (ie 2d element of figure)
  • angulo-hedron = corner 2d patch (ie surface of sur-hedron)
  • glomo-hedron = bounded by glomohedrix (3d ball)
  • poly-hedron = many 2-patches with a closure rule
  • multi-hedron = many 2-patches without closure in mind (eg a net)
  • peri-hedron = perimeter formed of 2d patches

Note that to make these to higher dimensions, one replaces the /hedr/ with other stems. One can still use the original words, but these can be given a clear meaning in the PG, and are regarded as alternatives to it.

  • edge = sur-latron (surface 1-patch)
  • circle = glomo-latron (bounded by globe-shaped 1d patch)
  • sphere = glomohedron (globe shaped 2-patch as boundary)
  • vertex = sur-teelon (surface 0-d patch)

LATIN and GREEK
--------------------

The tradition of latin and greek names, and not mixing them, greatly limits what might be said, and tends to add unnecessary confusion to the subject.

The "professionalism" in using these words is more a fait that one really has not sat down and tried to derive a concise etymology in terms of the subject, and link words to existing meanings.

When one looks at the PG words, once one knows the root stems and idioms, there is a better than average chance of guessing the meaning of a new word. The great multitude of words is brought to heal by a concise vocabulary.

A word like "angulo-zett-on" is then "corner-7D-patch", that is a 7D patch that stands as a corner. Since the corner can be around or surronding, we have that it is a relation between a surface 7D patch (ie surzetton) and a different patch which is either completely contained on the surface of (around-cornering), or completely contained on the surface of (surround-cornering).

Another distinction is to be made between the stems, in that while both mean the same thing, only the first have the secondary meaning.

  • poly / multi : many ; poly also implies closure
  • surface / perimeter: boundary, the first is an density gradient and can be inside the figure.
  • around / surround : closing on, around is orthogonal to, surround is in the space of
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Postby bo198214 » Sat Jun 03, 2006 11:30 am

@Wendy

1. What the hell is a fabric and a patch? These words are by no means defined mathematical terms (though patch is one with a probably different meaning as you use it). So I wanted to look up it on the so well-promoted polygloss and they are not listed!

2. Whats about application and examples? It would really give a fresh note to your explanation when you would relate it to the current discussion. I.e. how would the here listed shapes named when using your conventions. Or if you could provide any help with it in the naming of the toratopes/rotatopes etc.

3. I want to have (at least one) relevant example (i.e. a proposotion) where the application of your here introduced words/system is useful. I.e. makes a more concise description than in ordinary language. It is damn annoying to read a text where half the words have to be looked up in the authors personal language (might be that conlanger find it interesting) if it has no striking effects in clarity and simplicity!

4. the th-character in the polygloss sucks.


wendy wrote:The tradition of latin and greek names, and not mixing them, greatly limits what might be said, and tends to add unnecessary confusion to the subject.


So please, where limits a consisting greek numbering (and that was the topic) what might be said, and adds confusion??? It is nonsense to state that. For me a consistent numbering simply feels better.

Summary: If you want the polygloos widely accepted, you must provide easy accessibility and striking benefits in terms of application. Otherwise its a dead language doomed to your own occupation.
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Postby Keiji » Sat Jun 03, 2006 11:49 am

houserichichi wrote:Bi and duo are both latin prefixes. The greek equivalents are di or dy. The actual greek numeral 2 is "duo" (or dyo or di) though but the prefix isn't.

Numerical equivalent = Greek prefix -- Latin prefix

1 = mono -- uni
2 = di/dy -- bi/duo
3 = tri -- tri
4 = tetra -- quadri/quart

If we insist on using "tetra" then you must stick with tri, di/dy, and mono for consistency. (think unilateral, bilateral, trilateral, quadrilateral versus monad, dyad, triad, tetrad)


Then we can call it a diframe. ;)
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Postby Nick » Sat Jun 03, 2006 2:27 pm

If an "H" means hollow, how do you know which position to put in? For example, is EHE the same as EEH?
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Postby Keiji » Sat Jun 03, 2006 3:43 pm

No, it isn't.

E = line segment
EH = hollow square
EHE = hollow cube with the top and bottom faces removed.

EE = solid square
EEH = hollow cube.
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Postby Nick » Sat Jun 03, 2006 4:23 pm

So, would HEE be a cube with the edges removed (but still leaving the endpoints of those edges)?
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