A Subset of Phi Space

​
{a, f}
{2,2}
{b, c}
{3,-1}
{d, e}
{2,2}
show some points
point size
1
2
3
show icosahedron
show half-diagonals
(black vectors)
Cartesian coordinates:
{2+3ϕ,-1+2ϕ,2+2ϕ}
(colored vectors)
icosahedral coordinates:
{2,0,2,0,1,2}
Phi space is the set
{a+bϕ,c+dϕ,e+fϕ},ϕ=
1+
5
2
,a,b,c∈
. The Zome construction system is based on phi space.
This Demonstration looks at the subset of phi space
X={a+bϕ,c+dϕ,e+fϕ},ϕ=
1+
5
2
,a,b,c∈,(a+f)2,(b+c)2,(d+e)2∈
, that is, where each of
a
and
f
,
b
and
c
, and
d
and
e
have the same parity. Then each element of
X
can be expressed as an integral linear combination of the six half-diagonals of the icosahedron.
The points shown have coefficients
a
,
b
, … at most 3 in absolute value. The convex hull of
X
is a triacontahedron.

External Links

Constructing the Regular Icosahedron
Exact Coordinates of Golden Rhombic Solids
Golden Ratio (Wolfram MathWorld)
Phi Space

Permanent Citation

Izidor Hafner
​
​"A Subset of Phi Space"​
​http://demonstrations.wolfram.com/ASubsetOfPhiSpace/​
​Wolfram Demonstrations Project​
​Published: March 28, 2013