Dodecahedron
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Dodecahedron
by papernuke » Tue Nov 27, 2007 2:38 am
is a dodecahedron a 4D figure? i searched it on the wiki and they had a spinning model and it was 3D. but i remember Dodecahedrons were 4D in a thread..
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by zero » Tue Nov 27, 2007 7:51 am
A polygon is a many-sided figure in two dimensions. Each of its sides is a one dimensional line segment, or edge.
A polyhedron is a many-sided figure in three dimensions. Each of its sides is a two dimensional polygon, or face.
A polychoron (or polycell) is a many-sided figure in four dimensions. Each of its sides is a three dimensional polyhedron, or cell.
I'm sure someone has made up names for higher dimensions, but you have to stop somewhere so I choose to stop there. Generally, these are all called polytopes regardless of their number of dimensions. So if you are paying attention to the part of each word after "poly," you will notice that a dodecahedron is three dimensional polytope with "dodeca"-many faces.
Does that help?
A polyhedron is a many-sided figure in three dimensions. Each of its sides is a two dimensional polygon, or face.
A polychoron (or polycell) is a many-sided figure in four dimensions. Each of its sides is a three dimensional polyhedron, or cell.
I'm sure someone has made up names for higher dimensions, but you have to stop somewhere so I choose to stop there. Generally, these are all called polytopes regardless of their number of dimensions. So if you are paying attention to the part of each word after "poly," you will notice that a dodecahedron is three dimensional polytope with "dodeca"-many faces.
Does that help?
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by papernuke » Wed Nov 28, 2007 2:50 am
A polyhedron is a many-sided figure in three dimensions. Each of its sides is a two dimensional polygon, or face.
-zero
so a dodecahedron is a 3D object with 12 faces?
5d = teron
6d = peton
7d+ follow SI
-Hayate
shouldnt it be
4D=teron
5D=peton
6D=hexon (or whatever)
7D+= follow SI
?
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by zero » Wed Nov 28, 2007 5:29 am
papernuke wrote:so a dodecahedron is a 3D object with 12 faces?
You've got it!
As for the terminology, think of it this way. It describes the dimensions of the sides. That's what the root words have to match up with -- the sides. So each n dimensional figure is named after its many sides, and the dimensions of those sides are n-1.
Does that help?
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by papernuke » Wed Nov 28, 2007 5:36 am
yah so all the sides of a 3D object is 2D, and so an n dimensional figures faces are n-1 dimension.. okay i get it.
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by wendy » Wed Nov 28, 2007 7:39 am
The manner of describing polytopes is to use a concept of a number of patches, eg polyhedron actually means 'many seats'. The same root is seen in cat.hedr.al = over.seat (the place where the bishop (episciple) sits. Episcile = epi (all) + scope (see).
In any case, there is some confusion over the established terminology in mathematics, because /poly.gon/ actually is 'many knee', with gon and knee being the same word. Various reformers have advocated some kind of consistancy in the termonology.
Since the PG is intended to go to many dimensions, it implements the notion of 'fabric of space', (words in -ix), and 'patches of such fabric' (in -on.). So hedrIX is the fabric of 2-space, and hedr-ON is a specific patch.
The prefix applied to /hedron/ is then something one does with said patches. The usual pattern is to apply a numerical prefix, with the intent that such closes the figure. So a tetra.hedron is 4 2-patches that make a closed figure.
One can use other descriptive words to say an effect, rather than a count. 'apeiro' means 'without a perimeter (in space)' a = without (cf a.tom = without division), peri or peiro = boundary (cf perimeter). An a.peiro.hedron is patches laid in a plane without a perimeter: a 2d tiling.
One then distinguishes between /poly/ = many + closure, against just /multi/ = many. A miltihedron is not nessecarily closed, a partial polyhedron for example.
Likewise, 'hedrix' is used for 2d cloth, or manifolds. With suitable prefixes, it takes on assorted shapes too. A plato.hedr.ix is a flat+2d+coth (ie a plane). A glomo.hedrix = round+2d+cloth = sphere. A glomohedron is the corresponding sphere with interior.
By replacing /hedr/ with 3d /chor/, 4d /ter/, 5d /pet/, 6d /ect/, 7d /zett/ and 8d /yott/, one has ready-made names for figures bounded by these fabrics or patches. We for example, live in a horo.chor.ix, = horizon-centred + 3d + fabric. Hyperbolic 4-space would be a bollo.ter.ix. And so forth.
The actual metric prefix for E18 is 'Exa' greek /hexa/. However, the corresponding cloth would be *exix, which mutates under euphonics to 'ectix'. Since /ex/ is already an active prefix meaning "out of, former", it was decided to backform 'ect' throughout the whole of six dimensions.
In any case, there is some confusion over the established terminology in mathematics, because /poly.gon/ actually is 'many knee', with gon and knee being the same word. Various reformers have advocated some kind of consistancy in the termonology.
Since the PG is intended to go to many dimensions, it implements the notion of 'fabric of space', (words in -ix), and 'patches of such fabric' (in -on.). So hedrIX is the fabric of 2-space, and hedr-ON is a specific patch.
The prefix applied to /hedron/ is then something one does with said patches. The usual pattern is to apply a numerical prefix, with the intent that such closes the figure. So a tetra.hedron is 4 2-patches that make a closed figure.
One can use other descriptive words to say an effect, rather than a count. 'apeiro' means 'without a perimeter (in space)' a = without (cf a.tom = without division), peri or peiro = boundary (cf perimeter). An a.peiro.hedron is patches laid in a plane without a perimeter: a 2d tiling.
One then distinguishes between /poly/ = many + closure, against just /multi/ = many. A miltihedron is not nessecarily closed, a partial polyhedron for example.
Likewise, 'hedrix' is used for 2d cloth, or manifolds. With suitable prefixes, it takes on assorted shapes too. A plato.hedr.ix is a flat+2d+coth (ie a plane). A glomo.hedrix = round+2d+cloth = sphere. A glomohedron is the corresponding sphere with interior.
By replacing /hedr/ with 3d /chor/, 4d /ter/, 5d /pet/, 6d /ect/, 7d /zett/ and 8d /yott/, one has ready-made names for figures bounded by these fabrics or patches. We for example, live in a horo.chor.ix, = horizon-centred + 3d + fabric. Hyperbolic 4-space would be a bollo.ter.ix. And so forth.
The actual metric prefix for E18 is 'Exa' greek /hexa/. However, the corresponding cloth would be *exix, which mutates under euphonics to 'ectix'. Since /ex/ is already an active prefix meaning "out of, former", it was decided to backform 'ect' throughout the whole of six dimensions.
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by Keiji » Wed Nov 28, 2007 8:11 am
papernuke wrote:5d = teron
6d = peton
7d+ follow SI
-Hayate
shouldnt it be
4D=teron
5D=peton
6D=hexon (or whatever)
7D+= follow SI
?
No, I know what I'm talking about -_- 4D is choron, as zero said, so I didn't feel the need to duplicate.
A polychoron (or polycell) is a many-sided figure in four dimensions. Each of its sides is a three dimensional polyhedron, or cell.
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by papernuke » Thu Nov 29, 2007 3:08 am
Hayate wrote:5d = teron
6d = peton
7d+ follow SI
Then is it one dimension down for each dimension because its describing the faces? not the entire figure?
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by Keiji » Thu Nov 29, 2007 5:48 am
No, it's not one dimension down for any reason. 
a shape in 2d is a POLYGON
a shape in 3d is a POLYHEDRON
a shape in 4d is a POLYCHORON
a shape in 5d is a POLYTERON
oh look, I'm right. Learn to count.
a shape in 2d is a POLYGON
a shape in 3d is a POLYHEDRON
a shape in 4d is a POLYCHORON
a shape in 5d is a POLYTERON
oh look, I'm right. Learn to count.
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