The disphenocingulum is an icositetrahedron with 20 congruent equilateral triangles and 4 congruent squares as surfaces, 16 corners and 38 edges. Four edges adjoin four of the corners and five edges adjoin the other twelve corners. There are two edges in the body, each connecting 2 squares. These two edges are orthogonal and skewed .
It is the Johnson body J 90 from a series of 92 bodies named after the mathematician Norman Johnson .
Cartesian coordinates The Cartesian coordinates of the corner points can be, with center at origin and edge length 2:
(
0
,
±
1
,
p
)
(
±
s
,
±
1
,
q
)
(
0
,
±
t
,
r
)
(
±
t
,
0
,
-
r
)
(
±
1
,
±
s
,
-
q
)
(
±
1
,
0
,
-
p
)
{\ displaystyle {\ begin {alignedat} {4} & (& 0, && \; \ pm 1, && \; p &) \\ & (& \ pm s, && \; \ pm 1, && \; q &) \ \ & (& 0, && \; \ pm t, && \; r &) \\ & (& \ pm t, && \; 0, && \; - r &) \\ & (& \ pm 1, && \; \ pm s, && \; - q &) \\ & (& \ pm 1, && \; 0, && \; - p &) \ end {alignedat}}}
According to the Pythagorean theorem:
(
p
-
q
)
2
+
s
2
=
4th
{\ displaystyle (pq) ^ {2} + s ^ {2} = 4}
(
p
-
r
)
2
+
(
t
-
1
)
2
=
4th
{\ displaystyle (pr) ^ {2} + (t-1) ^ {2} = 4}
4th
q
2
+
s
2
+
(
t
-
s
)
2
=
4th
{\ displaystyle 4q ^ {2} + s ^ {2} + (ts) ^ {2} = 4}
2
q
2
+
(
s
-
1
)
2
=
2
{\ displaystyle 2q ^ {2} + (s-1) ^ {2} = 2}
(
q
+
r
)
2
+
(
t
-
s
)
2
=
3
{\ displaystyle (q + r) ^ {2} + (ts) ^ {2} = 3}
With
p
=
a
+
b
2
≈
2
,
21st
q
=
a
+
b
2
-
2
a
≈
0
,
93
r
=
b
-
a
2
≈
0
,
65
s
=
2
a
a
2
-
1
≈
1
,
53
t
=
4th
-
a
2
+
1
≈
2
,
25th
{\ displaystyle {\ begin {aligned} p & = {\ frac {a + b} {2}} \ approx 2 {,} 21 \\ q & = {\ frac {a + b} {2}} - {\ frac {2} {a}} \ approx 0 {,} 93 \\ r & = {\ frac {ba} {2}} \ approx 0 {,} 65 \\ s & = {\ frac {2} {a}} { \ sqrt {a ^ {2} -1}} \ approx 1 {,} 53 \\ t & = {\ sqrt {4-a ^ {2}}} + 1 \ approx 2 {,} 25 \ end {aligned} }}
consisting of
b
=
2
a
+
3
-
(
4th
-
a
2
+
1
-
2
a
a
2
-
1
)
2
≈
2
,
86
{\ displaystyle b = {\ frac {2} {a}} + {\ sqrt {3- \ left ({\ sqrt {4-a ^ {2}}} + 1 - {\ tfrac {2} {a} } {\ sqrt {a ^ {2} -1}} \ right) ^ {2}}} \ approx 2 {,} 86}
a
≈
1.558
86993226
{\ displaystyle a \ approx 1 {,} 55886993226}
4th
=
2
(
2
a
a
2
-
1
-
1
)
2
+
(
3
-
(
4th
-
a
2
-
2
a
a
2
-
1
+
1
)
2
+
a
-
2
a
)
2
{\ displaystyle 4 = 2 \ left ({\ frac {2} {a}} {\ sqrt {a ^ {2} -1}} - 1 \ right) ^ {2} + \ left ({\ sqrt {3 - \ left ({\ sqrt {4-a ^ {2}}} - {\ tfrac {2} {a}} {\ sqrt {a ^ {2} -1}} + 1 \ right) ^ {2} }} + a - {\ frac {2} {a}} \ right) ^ {2}}
Formulas
Sizes of a disphenocingulum with edge length a
Surface area
A.
O
=
a
2
(
4th
+
5
3
)
{\ displaystyle A_ {O} = a ^ {2} \ left (4 + 5 {\ sqrt {3}} \ right)}
Web links
<img src="https://de.wikipedia.org//de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
