Changes: Lunar-holed Rhombicosidodecahedron E5/6J91(P4)

Revision as of 12:47, 15 January 2008

FEED ME SEYMOUR!!!
The Lunar-holed Rhombicosidodecahedron, Stewart's E₅/6J₉₁(P₄)

Type	Stewart Toroid
Panels	20 × 24 × 36 ×
Rods	108 ×
Spheres	56 ×
Author	PolyClare 19:06, 14 January 2008 (UTC)

The Lunar-holed Rhombicosidodecahedron, Stewart's E₅/6J₉₁(P₄).

This is a Rhombicosidodecahedron (called E₅ by Johnson), with a 6-way hole through it made from Stewart's 6J₉₁(P₄), as described on pages 128-129 of Stewart's book (2nd ed.).

Stewart writes

It is unusual becuse each of the six openings is a lunar complex, rather than a single face, giving to the tunnels a novel, slit-like appearance.

By "lunar complex", Stewart is referring to a "lune", which Johnson defines to be:

WARNING: The pentagons only barely almost-fit in the outer shell of this model. That's because you can't comfortably fit a Geomag pentagon panel into the base of a pentagonal pyramid -- the angles are too extreme. For the pictures I took, I only put in the 4 visible pentagons, and then didn't breathe too heavily as I took the pictures, lest they pop out. If you look carefully at the pictures, you may be able to see that the pentagons stick out more than they should.

Counts of Rods, Spheres, Panels

From the page on Stewart's 6J₉₁(P₄), The totals were:

24 ×

6 ×

24 ×

120 ×

56 ×

The inner portion of a lune (inside the 6 outer rods and 6 spheres) is:

2 ×

1 ×

2 ×

Subtracting 6 inner lune portions from the totals above gives the "6J₉₁(P₄) Hole Subtotal":

12 ×

0 ×

24 ×

108 ×

56 ×

Now let's add the "alpha" parts (see Image #3 below) . We need to add just "top part" of the alpha: the clear panels and silver rods and central three spheres for each alpha. Each alpha resides above three pentagons of the 6J₉₁(P₄), and shares those pentagons with no other alpha. An alpha top part contains:

1 ×

3 ×

9 ×

3 ×

How many alpha top parts need to be added? Each alpha separately covers 3 pentagons, and the 6J₉₁(P₄) has 24 pentagons. Therefore 24/3 = 8 alpha top parts must be required.

Additionally, 12 pentagons (representing the pentagonal pyramid in Image #3 below) must be added.

Therefore, the overall total must be the "6J₉₁(P₄) Hole Subtotal" plus 8 alpha top parts and plus 12 pentagons:

20 ×

24 ×

36 ×

180 ×

80 ×

Other views

Notes to come soon -- in the meantime, here are the pictures:

@@ Line 1: / Line 1: @@
 {{Geomag-construction|PageTitle=Lunar-holed Rhombicosidodecahedron E5/6J91(P4)|Title=FEED ME SEYMOUR!!!|Filename=E5-6J91(P4) View 9 Holes Open .jpg|Caption=The Lunar-holed Rhombicosidodecahedron, Stewart's E<SUB>5</SUB>/6J<SUB>91</SUB>(P<SUB>4</SUB>)
 |Type=[[:Category:Stewart Toroid|Stewart Toroid]]
-|Triangles=|Squares=|Pentagons=|Rhombic=|Rods=|Spheres=|Author=[[User:PolyClare|PolyClare]] 19:06, 14 January 2008 (UTC)}}
+|Triangles=20|Squares=24|Pentagons=36|Rhombic=|Rods=108|Spheres=56|Author=[[User:PolyClare|PolyClare]] 19:06, 14 January 2008 (UTC)}}
 The Lunar-holed Rhombicosidodecahedron, Stewart's E<SUB>5</SUB>/6J<SUB>91</SUB>(P<SUB>4</SUB>).
@@ Line 13: / Line 13: @@
 <B>WARNING:</B> The pentagons only <I>barely</I> almost-fit in the outer shell of this model.  That's because you can't comfortably fit a Geomag pentagon panel into the base of a pentagonal pyramid -- the angles are too extreme.  For the pictures I took, I only put in the 4 visible pentagons, and then didn't breathe too heavily as I took the pictures, lest they pop out.  If you look carefully at the pictures, you may be able to see that the pentagons stick out more than they should.
+== Counts of Rods, Spheres, Panels ==
-<FONT COLOR=red><B>Rods/Spheres/Panels counts to come soon.</B></FONT>
+From the page on <B>[[Stewart's 6J91(P4)|Stewart's 6J<SUB>91</SUB>(P<SUB>4</SUB>)]]</B>, The totals were:
+<TABLE CELLPADDING=12>
+<TR><TD>  24 × [[Image:Geomag triangle panel icon.gif]]  </TD>
+    <TD>   6 × [[Image:Geomag square panel icon.gif]]    </TD>
+    <TD>  24 × [[Image:Geomag pentagon panel icon.gif]]  </TD>
+    <TD> 120 × [[Image:Geomag rod icon.gif]]             </TD>
+    <TD>  56 × [[Image:Geomag ball icon.gif]]            </TD>
+</TR></TABLE>
+The inner portion of a lune (inside the 6 outer rods and 6 spheres) is:
+<TABLE CELLPADDING=12>
+<TR><TD>   2 × [[Image:Geomag triangle panel icon.gif]]  </TD>
+    <TD>   1 × [[Image:Geomag square panel icon.gif]]    </TD>
+    <TD>   2 × [[Image:Geomag rod icon.gif]]             </TD>
+</TR></TABLE>
+Subtracting 6 inner lune portions from the totals above gives the "6J<SUB>91</SUB>(P<SUB>4</SUB>) Hole Subtotal":
+<TABLE CELLPADDING=12>
+<TR><TD>  12 × [[Image:Geomag triangle panel icon.gif]]  </TD>
+    <TD>   0 × [[Image:Geomag square panel icon.gif]]    </TD>
+    <TD>  24 × [[Image:Geomag pentagon panel icon.gif]]  </TD>
+    <TD> 108 × [[Image:Geomag rod icon.gif]]             </TD>
+    <TD>  56 × [[Image:Geomag ball icon.gif]]            </TD>
+</TR></TABLE>
+Now let's add the "<I>alpha</I>" parts (see [[:Image:E5-6J91(P4) View 3 Bits .jpg|Image #3 below]]) .  We need to add just "top part" of the <I>alpha</I>: the clear panels and silver rods and central three spheres for each <I>alpha</I>.  Each <I>alpha</I> resides above three pentagons of the 6J<SUB>91</SUB>(P<SUB>4</SUB>), and shares those pentagons with no other <I>alpha</I>.  An <I>alpha</I> top part contains:
+<TABLE CELLPADDING=12>
+<TR><TD>   1 × [[Image:Geomag triangle panel icon.gif]]  </TD>
+    <TD>   3 × [[Image:Geomag square panel icon.gif]]    </TD>
+    <TD>   9 × [[Image:Geomag rod icon.gif]]             </TD>
+    <TD>   3 × [[Image:Geomag ball icon.gif]]            </TD>
+</TR></TABLE>
+How many <I>alpha</I> top parts need to be added?  Each <I>alpha</I> separately covers 3 pentagons, and the 6J<SUB>91</SUB>(P<SUB>4</SUB>) has 24 pentagons.  Therefore 24/3 = 8 <I>alpha</I> top parts must be required.
+Additionally, 12 pentagons (representing the pentagonal pyramid in [[:Image:E5-6J91(P4) View 3 Bits .jpg|Image #3 below]]) must be added.
+Therefore, the overall total must be the "6J<SUB>91</SUB>(P<SUB>4</SUB>) Hole Subtotal" plus 8 <I>alpha</I> top parts and plus 12 pentagons:
+<TABLE CELLPADDING=12>
+<TR><TD>  20 × [[Image:Geomag triangle panel icon.gif]]  </TD>
+    <TD>  24 × [[Image:Geomag square panel icon.gif]]    </TD>
+    <TD>  36 × [[Image:Geomag pentagon panel icon.gif]]  </TD>
+    <TD> 180 × [[Image:Geomag rod icon.gif]]             </TD>
+    <TD>  80 × [[Image:Geomag ball icon.gif]]            </TD>
+</TR></TABLE>
 {{clr}}
@@ Line 26: / Line 72: @@
 <gallery>
-Image:E5-6J91(P4) View 3 Bits .jpg|3.
+Image:E5-6J91(P4) View 3 Bits .jpg|3. Name the larger piece "<I>alpha</I>".
 Image:E5-6J91(P4) View 4 Start .jpg|4.
 Image:E5-6J91(P4) View 5 Filled In .jpg|5.

Changes: Lunar-holed Rhombicosidodecahedron E5/6J91(P4)

Revision as of 12:47, 15 January 2008

Counts of Rods, Spheres, Panels

Other views

Related links