Biomedical Engineering Reference
In-Depth Information
D
A
A
′
D
′
A
′
y
B
′
a
F
y
E
′
F
′
B
C
y
O
r
q
C
′
y
B
′
C
′
A
c
A
″
D
′
E
C
P
D
a
(a)
B
(b)
(c)
Figure 1.37
(a) A tetrahedron has four
faces, four vertices, and six edges. A regular
tetrahedron is composed of four equilat-
eral triangular faces. All edges are equal:
AB
=
BC
=
AC
=
AD
=
BD
=
CD
=
a
.The
height of the triangular face, for example,
EC
=
√
3
a
/
2, the height of tetrahedron:
PD
=
√
2
/
3
a
, the distance between perpen-
dicular edges, for example, AB and CD is
equal to EF
=
a
/
√
2. (b) Top view of two
perpendicular tetrahedron edges: AD and
BC. In order to inscribe the tetrahedron in
a cylinder of radius
r
, the edges should be
turned with respect to the axis of the cylin-
der through an angle of
ψ
.Thedistance
a
/
√
2 between the perpendicular edges AD
and BC remains unchanged. (c) Projection
of tetrahedron edges on the cross section
of the cylinder circumscribed on the tetra-
hedron. The segment A
A
=
a
/
√
2(thedis-
tance between perpendicular edges AD and
BC). The length scale between figures (a),
(b), and (c) is not preserved.
According to Figure 1.37, we have projections A
D
=
and B
C
=
a
sin
ψ
a
cos
ψ
.
Hence, B
A
=
). Further (A
A
)
2
(B
A
)
2
(A
B
)
2
.Subse-
(
a
/
2)(sin
ψ
+
cos
ψ
+
=
/
√
10 and A
B
=
/
√
10
=
B
C
=
C
D
=
quently, cos
ψ
=
3
c
=
3
a
0
.
9486
a
.More-
/
√
10. Because A
D
=
over, A
D
=
2) and B
C
=
a
2
r
sin ((3
θ
−
2
π
)
/
2
r
sin (
θ/
2),
=
√
5
we get the angle
θ
and radius
r
of the cylinder. We get sin (
θ/
2)
/
6, cos
θ
=
3
√
3
a
=
√
3
3. Thus
∠
A
OB
=
∠
B
OC
=
∠
C
OD
=
θ
−
2
/
and
r
=
/
10
/
10
c
.
)
≈
131
.
81
◦
.
The angle
θ
is an irrational number, and the tetrahelix has no period. It belongs to
aperiodic crystals (according to Schrodinger [3]) or quasicrystals (according to the
contemporary notion).
Let the edge of the tetrahedron be equal to 1. The whole structure of a B-C helix
is generated by a screw transformation
x
→
R
x
+
a
where
The spiral moves along the tetrahelix by an angle
θ
=
arccos
(
−
2
/
3
3
√
2
1
/
3
√
2
1
/
3
√
2
221
2
1
/
1
3
R
=
,
a
=
2
−
12
−
2
−
1
−
Sadoc and Charvolin have indicated that the idea of three-sphere fibrations may be a
tool for analyzing twisted materials in condensed matter. They indicated that chiral
molecules, when densely packed in soft condensed matter or biological materials,
build organizations that are most often spontaneously twisted. The formation of
these organizations is driven by the fact that compactness, which tends to align the
