Previous articles in this series:
◼
  • Squeezing Pi from a Menger Sponge
    • ◼
  • Volume=3.141 -- The Canonical Tetragonal Antiwedge Hexahedron
    • It’s Pi-day soon, so it’s good to bring out odd things about Pi. For example, here’s a high-power approximation:
    In[]:=
    NPi-
    7
    7
    9
    4
    
    Out[]=
    0.0000254234
    There are other interesting constants, such at the golden ratio:
    In[]:=
    Solve[ϕ^2==ϕ+1]
    Out[]=
    ϕ
    1
    2
    (1-
    5
    ),ϕ
    1
    2
    (1+
    5
    )
    If coins have a golden ratio to each other, three golden coins will surround a unit coin against a wall. The penny and half dollar are close to this ratio.
    In[]:=
    ϕ=GoldenRatio;​​GraphicsLine[{{-Pi,0},{Pi,0}}],Circle[{0,1},1],Circle[{0,2+ϕ},ϕ],Circle2
    ϕ
    ,ϕ,ϕ,Circle-2
    ϕ
    ,ϕ,ϕ
    Out[]=
    Another is the tribonacci constant,
    3
    t
    -
    2
    t
    -t-1=0
    , seen in the snub cube. The snub dodecahedron needs a root of
    3
    t
    -
    2
    t
    -t-ϕ=0
    .
    In[]:=
    scroot=Root[-1-#1-
    2
    #1
    +
    3
    #1
    &,1];​​{scv,scf}=
    ;scp=ResourceFunction["SqrtSpace"][scroot,scv];​​sdroot=Root[-GoldenRatio-#1-
    2
    #1
    +
    3
    #1
    &,1];​​{sdv,sdf}=
    ;sdp=ResourceFunction["SqrtSpace"][sdroot,sdv];​​GraphicsRow[{​​Graphics3D[{EdgeForm[Thick],Opacity[.9],Polygon[scp[[#]]]&/@scf},BoxedFalse,ViewAnglePi/9],Graphics3D[{EdgeForm[Thick],Opacity[.9],Polygon[sdp[[#]]]&/@sdf},BoxedFalse,ViewAnglePi/10]},ImageSize530]
    Out[]=
    When set in terms of these roots, the complicated vertices are simple integers:
    In[]:=
    Union[Abs[Flatten[{scv,sdv}]]]
    Out[]=
    {0,1,2,3,4,5,6,7,8,9,10,11,12,13}
    Five coins can surround a unit coin against a wall if they have a tribonacci-based ratio:
    In[]:=
    t2=1
    1.84
    …
    ^2-2;GraphicsLine[{{-Pi,0},{Pi,0}}],Circle[{0,1},1],Circle[{0,2+t2},t2],Circle2
    t2
    ,t2,t2,Circle-2
    t2
    ,t2,t2,Circle
    1.31
    …
    ,
    2.12
    …
    ,t2,​​Circle-
    1.31
    …
    ,
    2.12
    …
    ,t2
    Out[]=
    Here’s a little trick done with e, π and
    19
    .
    In[]:=
    N[(E^(Sqrt[19]Pi)+24)^(1/24),11]
    Out[]=
    1.7692923542
    That seems curiously close to the root of
    3
    x
    =2(x+1)
    :
    In[]:=
    N[Root[-2-2#1+
    3
    #1
    &,1],11]
    Out[]=
    1.7692923542
    We can use that root for the optimal covering of a circle by twelve circles:
    Out[]=
    Melissen's 12 Disk Covering
    Here’s a similar trick done with e, π and
    163
    .
    In[]:=
    N[(E^(Sqrt[163]Pi)+24)^(1/24),33]
    Out[]=
    5.31862821775018565910968015331802
    That seems curiously close to the root of
    3
    x
    =2(3
    2
    x
    -2x+1)
    :
    In[]:=
    N[Root[-2+4#1-6#1^2+
    3
    #1
    &,1],33]
    Out[]=
    5.31862821775018565910968015331802
    For an explanation, see my column Shattering the Plane. You might also like The Circles of Descartes.

    CITE THIS NOTEBOOK

    Pi, the Golden Ratio and the Tribonacci Constant​
    by Ed Pegg​
    Wolfram Community, STAFF PICKS, March 13, 2024
    ​https://community.wolfram.com/groups/-/m/t/3140342