EihiS

October 2, 2014

Dodecahedron construction starting from a simple hexagon (cube)

Filed under: Eih geometry — admin @ 8:30 am

Construction : Start from an hexagon.

Pentagon’s circle is constructed from line draw between point J and I (green line) , intersecting the hexagon’s circle C at point K (center is I ) - plane 2D pentagon construction is detailed later below

  • circumference C / circumference P gives ratio 1.86834
  • segment’s length CG / LM gives ratio 1.58930
  • segment CG/ segment LK gives 0.98224
Correlations Notes :
From “ildenir”’s 3d models files, on github , dodécahédron’s 20 vertices coordinates (x,y,z) :

1.21412,0,1.58931,
0.375185,1.1547,1.58931,
-0.982247,0.713644,1.58931,
-0.982247,-0.713644,1.58931,
0.375185,-1.1547,1.58931,
1.96449,0,0.375185,
0.607062,1.86834,0.375185,
-1.58931,1.1547,0.375185,
-1.58931,-1.1547,0.375185,
0.607062,-1.86834,0.375185,
1.58931,1.1547,-0.375185,
-0.607062,1.86834,-0.375185,
-1.96449,0,-0.375185,
-0.607062,-1.86834,-0.375185,
1.58931,-1.1547,-0.375185,
0.982247,0.713644,-1.58931,
-0.375185,1.1547,-1.58931,
-1.21412,0,-1.58931,
-0.375185,-1.1547,-1.58931,
0.982247,-0.713644,-1.58931

dodecahedron from simple hexagon 2D ortho :

  • STEP1 : drawing the simple hexagon figure
  • STEP2 : using the PHI / Pentagon found point to have some coordinates of the dodecahedron’s figure we want to construct:
    create line K-JJ.
    JJ is The Point
    Draw circle E-JJ : EL is the distance for dodecahedron draw
  • STEP3 : using EL all around the circle:
    We now have L + M,N,O,P,Q = 6 points more
  • STEP4 :
    segments are created from L,M,N,O,P,Q to the hexagon’s original points on the circle
  • STEP5:
    Draw parallels l1,m1,n1. it intersects the hexagon’s axes at S,R,T points
  • STEP6-7-8-9:
    from R,S,T, construct the dodecahedron’s base pentagon’s edges and the rest of segments :
_________________________
Simplest pentagon construction, starting from an hexagon figure :
Hexagon has a starting point. line LK is then created. M is the intersection of this line with the base circle. then, draw a circle with center K, through M. this circle intersects the vertical center axis of the hexagon, at points N and O. drawing the lines M-N , and M-O gives the base angle of the pentagon. ANGLE of NMO is 36°

Hexagon has a starting point. line LK is then created. M is the intersection of this line with the base circle. then, draw a circle with center K, through M. this circle intersects the vertical center axis of the hexagon, at points N and O. drawing the lines M-N , and M-O gives the base angle of the pentagon. ANGLE of NMO is 36°

—————

dodecahedron view from Z, starting from the previous figure :

The outside delimiting circle has center K thru C ,L ). Then the dodecahedron's pentagons can be draw from the first element's axes and projections

The outside delimiting circle has center K thru C ,L ). Then the dodecahedron's pentagons can be draw from the first element's axes and projections

314159265358979323846264338327950288
419716939937510582097494459230781640
628620899862803482534211706798214808

cat{ 174 } { post_645 } { } 2009-2015 EIhIS Powered by WordPress