Plexagon - Pleated Hexagon

(hexagonal dipyramidal scanelohedra)

Written by Paul Bourke
Based upon concepts/work by Ron Evans
February 2000


Quote

A new modular building geometry--Plexagon is a new modular geometry based on the radial pleating of polygonal plates, and their joining or co-bracing into 3D modules. This engineering approach can form a very practical basis for modular greenhouses (using recycled plastic sheeting) and modular homes (using fibrous and metal sheeting) for the world's homeless and refugees. The word Plexagon is an abbreviation of "pleated hexagon". Plexagon is a newly-identified family of hexagonal dipyramidal scanelohedra.
Ron Evans

Construction

The construction of a plexagon is illustrated below. Start with two hexagons with radius and edge length "n", extend every second radial length to "i", fold the radial edges in the appropriate direction, and finally join the two halves together.

  "Bottom" face "Top" face
Hexagon
(2D)
Extend
alternate
vertices
(2D)
  E2 = n2 - n i + i2  
Fold
edges
above
or
below
plane
(2D)
Join
"top"
and
"bottom"
sheets
(3D view)

Animation illustrating plexagon formation

Templates for making your own plexagon

The instructions above make it easy to measure and cut the top and bottom plexagon pieces from stiff paper. To make it easier for two specific plexagons the following two files can be printed, they represent the T90 and T120 plexagons. You should be able to print these directly from your browser, they will fit onto both A4 and US-Letter paper sizes.

Vertex and edge numbering conventions

Numbering conventions for vertices and edges in the computer model.

Panels
n = 1.0, i = 1.5

DXF file
PovRay model
Vision3d text file

Panel

Additional tetrahedra to fill in corners
n = 1.0, i = 1.2

DXF file
PovRay model
Vision3d text file

Corner

T90 square enclosure
n = 1.01311, i = 1.19952

DXF file
PovRay model
Vision3d text file

T90 rectangle

Two plexagon panels can be joined by taking the "sawtooth" edge of one, reversing it, and joining it at some angle to the "sawtooth" edge of another panel. The angle (T) the panels need to be rotated by in order to form a perfect seal depends upon the ratio of "i" and "n". This is illustrated on the right.

As the ratio i/n (pleat ratio) is increased the rotation angle T decreases. T is at a maximum (180) as i approaches n (a flat sheet). As i/n increases the plexagon panels will eventually intersect each other at about i/n equal to 1.31.

Animation illustrating T angle

Table of n, i, E, and T (degrees)

    n           i           E          T
========    ========    ========    ==========
1.000000    1.866760    Golden ratio             dxf, pov, vision3d
1.000000    1.700000    1.479865     46.233918
1.000000    1.690000    1.471768     46.659752
1.000000    1.680000    1.463694     47.093682
1.000000    1.670000    1.455644     47.535952
1.000000    1.660000    1.447619     47.986815
1.000000    1.650000    1.439618     48.446535
1.000000    1.640000    1.431642     48.915387
1.000000    1.630000    1.423692     49.393660
1.000000    1.620000    1.415768     49.881654
1.000000    1.610000    1.407871     50.379684
1.000000    1.600000    1.400000     50.888078   dxf, pov, vision3d
1.000000    1.590000    1.392157     51.407180
1.000000    1.580000    1.384341     51.937350
1.000000    1.570000    1.376554     52.478966
1.000000    1.560000    1.368795     53.032424
1.000000    1.550000    1.361066     53.598140
1.000000    1.540000    1.353366     54.176549
1.000000    1.530000    1.345697     54.768111
1.000000    1.520000    1.338058     55.373307
1.000000    1.510000    1.330451     55.992646
1.000000    1.500000    1.322876     56.626663
1.000000    1.490000    1.315333     57.275922
1.000000    1.480000    1.307823     57.941019
1.000000    1.470000    1.300346     58.622583
1.000000    1.460000    1.292904     59.321280
1.000000    1.450520    1.285880     60.000000   dxf, pov, vision3d
1.000000    1.450000    1.285496     60.037814
1.000000    1.440000    1.278124     60.772933
1.000000    1.430000    1.270787     61.527428
1.000000    1.420000    1.263487     62.302141
1.000000    1.410000    1.256225     63.097968
1.000000    1.400000    1.249000     63.915861 
1.000000    1.390000    1.241813     64.756839
1.000000    1.380000    1.234666     65.621985
1.000000    1.370000    1.227559     66.512462
1.000000    1.360000    1.220492     67.429513
1.000000    1.350000    1.213466     68.374472
1.000000    1.340000    1.206482     69.348774
1.000000    1.330000    1.199542     70.353962
1.000000    1.320000    1.192644     71.391702
1.000000    1.310000    1.185791     72.463794
1.000000    1.300000    1.178983     73.572190
1.000000    1.290000    1.172220     74.719007
1.000000    1.280000    1.165504     75.906551
1.000000    1.270000    1.158836     77.137338
1.000000    1.260000    1.152215     78.414123
1.000000    1.250000    1.145644     79.739929
1.000000    1.240000    1.139122     81.118086
1.000000    1.230000    1.132652     82.552277
1.000000    1.220000    1.126233     84.046586
1.000000    1.210000    1.119866     85.605564
1.000000    1.200000    1.113553     87.234305
1.000000    1.190000    1.107294     88.938542
1.000000    1.184000    1.103565     90.000000   dxf, pov, vision3d
1.000000    1.180000    1.101090     90.724759
1.000000    1.170000    1.094943     92.600334
1.000000    1.160000    1.088853     94.573726
1.000000    1.150000    1.082820     96.654702
1.000000    1.140000    1.076847     98.854642
1.000000    1.130000    1.070934    101.186934
1.000000    1.120000    1.065082    103.667510
1.000000    1.110000    1.059292    106.315585
1.000000    1.100000    1.053565    109.154694
1.000000    1.090000    1.047903    112.214204
1.000000    1.080000    1.042305    115.531598
1.000000    1.070000    1.036774    119.156064
1.000000    1.067800    1.035567    120.000000   dxf, pov, vision3d
1.000000    1.060000    1.031310    123.154476
1.000000    1.050000    1.025914    127.622055
1.000000    1.040000    1.020588    132.703231
1.000000    1.035924    1.018437    135.000000   dxf, pov, vision3d
1.000000    1.030000    1.015332    138.638092
1.000000    1.020000    1.010149    145.888603
1.000000    1.010000    1.005037    155.630794

The turning angle T defined in this way is the same as the interior angle if two plexagons are joined as shown in the diagram on the right. That is, joining two plexagons such that they share two "E" length edges, (there is only one way to do this).

Radius (R)

The plexagon radius R is the maximum radius of the plexagon if it is projected onto its radial plane. Note that this is not simply the length of the longest hexagonal rib because the ribs are angled in the dimension perpendicular to the radial plane. The value of R given a plexagon "i" and "n" can be found by solving the following:

E2 = R2 + [ sqrt(i2 - R2) - sqrt(n2 - R2) ]2

E was given earlier in terms of i and n as

E2 = n2 - n i + i2
Photographic contributin by Gayla Chandler