Tutorial Post 17: Constructing regular icosahedron.

In the tutorial post about folding, we constructed  regular tetrahedron. That was easy: we aligned two equilateral triangles to the sides of another one and folded them to obtain all the needed endpoints.

Now we show how to construct regular icosahedron.


Create two copies of equilateral triangles (as it was made in the folding post) and one outline of regular pentagon. Select the enpoints of the pentagon and set constructive points for scaling the pentagon so that its side will become as long as the side of the triangles.

Invoke scaling with the key sequence  'QMS1'.

Prepare for alignment of the first triangle to the side of pentagon with a three-points transformation. The constructive points may be set as on the following figure:

Invoke the three-points transformation (key sequence 'QMT').

The same way, align the second triangle to the adjacent side of the pentagon:

Now we have to prepare the folding. Fortunately, each triangle is in a distinct set, so we can use the 'N' key to select the set and key sequence 'QSS' to select (and then also bookmark) the traingle.
The constructive points for folding should be set as follows:

After folding ('QMF') you should have something like that:

I have also set light ('QDL') to have different shades on the triangles.

Using cursor jumping, insert the remaining sides of the pentagonal pyramid:

Now we could safely remove the segments, which are the outline of the pentagon, but we will leave them to make the placement of pryramids more visible in the remaining stages.

Now select everything ('QSX'). Extract the selected vertices to a single set ('QSE') and make three copies of the pyramids:

We still need one more triangle. We could have  made one more copy at the begining, but  we are adding it now:

We have to align two of  the pyramids with the triangle using three-points transformations:

Prepare for folding of the aligned pyramids:

After folding you get something like that:

We have to append the last pyramid to our construction. Prepare for three-points transformation:

And execute the transformation. The pyramid has been placed as on the view below:

The construction is almost ready. Remainig triangles can be inserted by cursor jumping to the existing endpoints:

To make the construction balanced we can select all endpoints ('QSX') and remove all segments, which are the outlines of the pentagons ('QD4').
Then

• Move the cursor to the centroid of the selected endpoins (which is now the center of the icosahedron) with key sequence 'QMJC'.
• Set the constructive point 'A' ('QPSA')
• Using INPUTS PAGE ('QI') set cursor's X,Y,Z coordinates to zeroes.
• Set the constructive point 'B' ('QPSB')
• Move the (still selected) endpoints by the vector 'AB' (key sequence 'QMM0')

Now we have icosahedron with the center in (0,0,0).

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