Does there exist a higher-dimensional 5-sided “tetrahedron + 1”?
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The first shape is "0"-sided and is a point. The next shape is a line segment and it's "1"-sided. The next shape is a triangle and it's 3-sided. The next shape is a tetrahedron and it's 4-sided. Can we define some higher-dimensional 5-sided regular shape which is the next shape in this sequence?
You can reach the next shape by placing a point equidistant to all previous points in the next higher dimension.
general-topology polyhedra natural-numbers polytopes simplicial-stuff
asked Dec 31 '18 at 19:37
I Said Roll Up n Smoke AdjointI Said Roll Up n Smoke Adjoint
9,33252659
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add a comment |
$begingroup$
The first shape is "0"-sided and is a point. The next shape is a line segment and it's "1"-sided. The next shape is a triangle and it's 3-sided. The next shape is a tetrahedron and it's 4-sided. Can we define some higher-dimensional 5-sided regular shape which is the next shape in this sequence?
You can reach the next shape by placing a point equidistant to all previous points in the next higher dimension.
general-topology polyhedra natural-numbers polytopes simplicial-stuff
asked Dec 31 '18 at 19:37
I Said Roll Up n Smoke AdjointI Said Roll Up n Smoke Adjoint
9,33252659
$endgroup$
8
$begingroup$
In general, this is called a "simplex" or "$n$-simplex".
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– Blue
Dec 31 '18 at 19:40
2
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@Blue thanks. I will read that
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– I Said Roll Up n Smoke Adjoint
Dec 31 '18 at 19:43
3
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In particular, see 5-cell. Here the 5 "sides" are tetrahedra, just like the 4 sides of a tetrahedron are triangles, the 3 sides of a triangle are line segments, and the 2 "sides" (or rather, ends) of a line segment are points.
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– Rahul
Dec 31 '18 at 20:02
2
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The 4D 24-cell though not a simplex, shares simplex-like features, being a regular polytope which is self dual.
$endgroup$
– Zachary Hunter
Dec 31 '18 at 23:17
add a comment |
$begingroup$
The first shape is "0"-sided and is a point. The next shape is a line segment and it's "1"-sided. The next shape is a triangle and it's 3-sided. The next shape is a tetrahedron and it's 4-sided. Can we define some higher-dimensional 5-sided regular shape which is the next shape in this sequence?
You can reach the next shape by placing a point equidistant to all previous points in the next higher dimension.
general-topology polyhedra natural-numbers polytopes simplicial-stuff
asked Dec 31 '18 at 19:37
I Said Roll Up n Smoke AdjointI Said Roll Up n Smoke Adjoint
9,33252659
$endgroup$
The first shape is "0"-sided and is a point. The next shape is a line segment and it's "1"-sided. The next shape is a triangle and it's 3-sided. The next shape is a tetrahedron and it's 4-sided. Can we define some higher-dimensional 5-sided regular shape which is the next shape in this sequence?
You can reach the next shape by placing a point equidistant to all previous points in the next higher dimension.
general-topology polyhedra natural-numbers polytopes simplicial-stuff
general-topology polyhedra natural-numbers polytopes simplicial-stuff
asked Dec 31 '18 at 19:37
I Said Roll Up n Smoke AdjointI Said Roll Up n Smoke Adjoint
9,33252659
asked Dec 31 '18 at 19:37
I Said Roll Up n Smoke AdjointI Said Roll Up n Smoke Adjoint
9,33252659
edited Dec 31 '18 at 19:40
I Said Roll Up n Smoke Adjoint
asked Dec 31 '18 at 19:37
I Said Roll Up n Smoke AdjointI Said Roll Up n Smoke Adjoint
9,33252659
asked Dec 31 '18 at 19:37
I Said Roll Up n Smoke AdjointI Said Roll Up n Smoke Adjoint
9,33252659
asked Dec 31 '18 at 19:37
I Said Roll Up n Smoke AdjointI Said Roll Up n Smoke Adjoint
9,33252659
9,33252659
8
$begingroup$
In general, this is called a "simplex" or "$n$-simplex".
$endgroup$
– Blue
Dec 31 '18 at 19:40
2
$begingroup$
@Blue thanks. I will read that
$endgroup$
– I Said Roll Up n Smoke Adjoint
Dec 31 '18 at 19:43
3
$begingroup$
In particular, see 5-cell. Here the 5 "sides" are tetrahedra, just like the 4 sides of a tetrahedron are triangles, the 3 sides of a triangle are line segments, and the 2 "sides" (or rather, ends) of a line segment are points.
$endgroup$
– Rahul
Dec 31 '18 at 20:02
2
$begingroup$
The 4D 24-cell though not a simplex, shares simplex-like features, being a regular polytope which is self dual.
$endgroup$
– Zachary Hunter
Dec 31 '18 at 23:17
add a comment |
8
$begingroup$
In general, this is called a "simplex" or "$n$-simplex".
$endgroup$
– Blue
Dec 31 '18 at 19:40
2
$begingroup$
@Blue thanks. I will read that
$endgroup$
– I Said Roll Up n Smoke Adjoint
Dec 31 '18 at 19:43
3
$begingroup$
In particular, see 5-cell. Here the 5 "sides" are tetrahedra, just like the 4 sides of a tetrahedron are triangles, the 3 sides of a triangle are line segments, and the 2 "sides" (or rather, ends) of a line segment are points.
$endgroup$
– Rahul
Dec 31 '18 at 20:02
2
$begingroup$
The 4D 24-cell though not a simplex, shares simplex-like features, being a regular polytope which is self dual.
$endgroup$
– Zachary Hunter
Dec 31 '18 at 23:17
8
8
$begingroup$
In general, this is called a "simplex" or "$n$-simplex".
$endgroup$
– Blue
Dec 31 '18 at 19:40
$begingroup$
In general, this is called a "simplex" or "$n$-simplex".
$endgroup$
– Blue
Dec 31 '18 at 19:40
2
2
$begingroup$
@Blue thanks. I will read that
$endgroup$
– I Said Roll Up n Smoke Adjoint
Dec 31 '18 at 19:43
$begingroup$
@Blue thanks. I will read that
$endgroup$
– I Said Roll Up n Smoke Adjoint
Dec 31 '18 at 19:43
3
3
$begingroup$
In particular, see 5-cell. Here the 5 "sides" are tetrahedra, just like the 4 sides of a tetrahedron are triangles, the 3 sides of a triangle are line segments, and the 2 "sides" (or rather, ends) of a line segment are points.
$endgroup$
– Rahul
Dec 31 '18 at 20:02
$begingroup$
In particular, see 5-cell. Here the 5 "sides" are tetrahedra, just like the 4 sides of a tetrahedron are triangles, the 3 sides of a triangle are line segments, and the 2 "sides" (or rather, ends) of a line segment are points.
$endgroup$
– Rahul
Dec 31 '18 at 20:02
2
2
$begingroup$
The 4D 24-cell though not a simplex, shares simplex-like features, being a regular polytope which is self dual.
$endgroup$
– Zachary Hunter
Dec 31 '18 at 23:17
$begingroup$
The 4D 24-cell though not a simplex, shares simplex-like features, being a regular polytope which is self dual.
$endgroup$
– Zachary Hunter
Dec 31 '18 at 23:17
add a comment |
1 Answer
1
active
oldest
votes
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The simplex is the easiest polytope, existing within all dimensions. It is nothing but the pyramid on a simplicial base (of one dimension less). In fact, take any simplex, attach to every facet a copy of that simplex again, and fold those copies up into the next dimension, then those will all meet at a single point atop the central simplex, which then will become its base.
This very construction already shows by induction, that the D-dimensional simplex would have D+1 facets, which all are D-1 dimensional simplices. Thus I can be affirmative, the 4th dimensional simplex is a pentachoron, i.e. it is built from 5 cells, all being tetrahedra.
In fact, the element count of all subelements of any dimension would be given by the numbers of the Pascal triangle: eg. a triangle has 3 vertices and 3 sides; a tetrahedron has 4 vertices, 6 edges, and 4 faces; a pentachoron has 5 vertices, 10 edges, 10 triangles, and 5 tetrahedra; etc.
The constructive device given above surely requires that the circumradius of the lower dimensional simplex is less than one edge unit, in order to allow to build an unit-edged pyramid on top. Again by induction, using right that very construction of a D-dimensional simplex, you could derive that crucial circumradius formula being
$$r=sqrt{frac{D}{2(D+1)}},$$
thus prooving its existance for every dimension.
--- rk
answered Jan 1 at 3:14
Dr. Richard KlitzingDr. Richard Klitzing
1,79526
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add a comment |
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$begingroup$
The simplex is the easiest polytope, existing within all dimensions. It is nothing but the pyramid on a simplicial base (of one dimension less). In fact, take any simplex, attach to every facet a copy of that simplex again, and fold those copies up into the next dimension, then those will all meet at a single point atop the central simplex, which then will become its base.
This very construction already shows by induction, that the D-dimensional simplex would have D+1 facets, which all are D-1 dimensional simplices. Thus I can be affirmative, the 4th dimensional simplex is a pentachoron, i.e. it is built from 5 cells, all being tetrahedra.
In fact, the element count of all subelements of any dimension would be given by the numbers of the Pascal triangle: eg. a triangle has 3 vertices and 3 sides; a tetrahedron has 4 vertices, 6 edges, and 4 faces; a pentachoron has 5 vertices, 10 edges, 10 triangles, and 5 tetrahedra; etc.
The constructive device given above surely requires that the circumradius of the lower dimensional simplex is less than one edge unit, in order to allow to build an unit-edged pyramid on top. Again by induction, using right that very construction of a D-dimensional simplex, you could derive that crucial circumradius formula being
$$r=sqrt{frac{D}{2(D+1)}},$$
thus prooving its existance for every dimension.
--- rk
answered Jan 1 at 3:14
Dr. Richard KlitzingDr. Richard Klitzing
1,79526
$endgroup$
add a comment |
$begingroup$
The simplex is the easiest polytope, existing within all dimensions. It is nothing but the pyramid on a simplicial base (of one dimension less). In fact, take any simplex, attach to every facet a copy of that simplex again, and fold those copies up into the next dimension, then those will all meet at a single point atop the central simplex, which then will become its base.
This very construction already shows by induction, that the D-dimensional simplex would have D+1 facets, which all are D-1 dimensional simplices. Thus I can be affirmative, the 4th dimensional simplex is a pentachoron, i.e. it is built from 5 cells, all being tetrahedra.
In fact, the element count of all subelements of any dimension would be given by the numbers of the Pascal triangle: eg. a triangle has 3 vertices and 3 sides; a tetrahedron has 4 vertices, 6 edges, and 4 faces; a pentachoron has 5 vertices, 10 edges, 10 triangles, and 5 tetrahedra; etc.
The constructive device given above surely requires that the circumradius of the lower dimensional simplex is less than one edge unit, in order to allow to build an unit-edged pyramid on top. Again by induction, using right that very construction of a D-dimensional simplex, you could derive that crucial circumradius formula being
$$r=sqrt{frac{D}{2(D+1)}},$$
thus prooving its existance for every dimension.
--- rk
answered Jan 1 at 3:14
Dr. Richard KlitzingDr. Richard Klitzing
1,79526
$endgroup$
add a comment |
$begingroup$
The simplex is the easiest polytope, existing within all dimensions. It is nothing but the pyramid on a simplicial base (of one dimension less). In fact, take any simplex, attach to every facet a copy of that simplex again, and fold those copies up into the next dimension, then those will all meet at a single point atop the central simplex, which then will become its base.
This very construction already shows by induction, that the D-dimensional simplex would have D+1 facets, which all are D-1 dimensional simplices. Thus I can be affirmative, the 4th dimensional simplex is a pentachoron, i.e. it is built from 5 cells, all being tetrahedra.
In fact, the element count of all subelements of any dimension would be given by the numbers of the Pascal triangle: eg. a triangle has 3 vertices and 3 sides; a tetrahedron has 4 vertices, 6 edges, and 4 faces; a pentachoron has 5 vertices, 10 edges, 10 triangles, and 5 tetrahedra; etc.
The constructive device given above surely requires that the circumradius of the lower dimensional simplex is less than one edge unit, in order to allow to build an unit-edged pyramid on top. Again by induction, using right that very construction of a D-dimensional simplex, you could derive that crucial circumradius formula being
$$r=sqrt{frac{D}{2(D+1)}},$$
thus prooving its existance for every dimension.
--- rk
answered Jan 1 at 3:14
Dr. Richard KlitzingDr. Richard Klitzing
1,79526
$endgroup$
The simplex is the easiest polytope, existing within all dimensions. It is nothing but the pyramid on a simplicial base (of one dimension less). In fact, take any simplex, attach to every facet a copy of that simplex again, and fold those copies up into the next dimension, then those will all meet at a single point atop the central simplex, which then will become its base.
This very construction already shows by induction, that the D-dimensional simplex would have D+1 facets, which all are D-1 dimensional simplices. Thus I can be affirmative, the 4th dimensional simplex is a pentachoron, i.e. it is built from 5 cells, all being tetrahedra.
In fact, the element count of all subelements of any dimension would be given by the numbers of the Pascal triangle: eg. a triangle has 3 vertices and 3 sides; a tetrahedron has 4 vertices, 6 edges, and 4 faces; a pentachoron has 5 vertices, 10 edges, 10 triangles, and 5 tetrahedra; etc.
The constructive device given above surely requires that the circumradius of the lower dimensional simplex is less than one edge unit, in order to allow to build an unit-edged pyramid on top. Again by induction, using right that very construction of a D-dimensional simplex, you could derive that crucial circumradius formula being
$$r=sqrt{frac{D}{2(D+1)}},$$
thus prooving its existance for every dimension.
--- rk
answered Jan 1 at 3:14
Dr. Richard KlitzingDr. Richard Klitzing
1,79526
answered Jan 1 at 3:14
Dr. Richard KlitzingDr. Richard Klitzing
1,79526
answered Jan 1 at 3:14
Dr. Richard KlitzingDr. Richard Klitzing
1,79526
answered Jan 1 at 3:14
Dr. Richard KlitzingDr. Richard Klitzing
1,79526
1,79526
add a comment |
add a comment |
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8
$begingroup$
In general, this is called a "simplex" or "$n$-simplex".
$endgroup$
– Blue
Dec 31 '18 at 19:40
2
$begingroup$
@Blue thanks. I will read that
$endgroup$
– I Said Roll Up n Smoke Adjoint
Dec 31 '18 at 19:43
3
$begingroup$
In particular, see 5-cell. Here the 5 "sides" are tetrahedra, just like the 4 sides of a tetrahedron are triangles, the 3 sides of a triangle are line segments, and the 2 "sides" (or rather, ends) of a line segment are points.
$endgroup$
– Rahul
Dec 31 '18 at 20:02
2
$begingroup$
The 4D 24-cell though not a simplex, shares simplex-like features, being a regular polytope which is self dual.
$endgroup$
– Zachary Hunter
Dec 31 '18 at 23:17