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Math Help - Please assist me in creating a Fibonacci Parallelogram

  1. #1
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    Please assist me in creating a Fibonacci Parallelogram

    Hi all,

    Again I am off on a project of my own related to design work I am doing.

    Please move if this is the wrong forum for this question - I suspect it is not really College Level geometry (well I certainly am not ;-) )

    Basically I want to reproduce a figure that I have grabbed a 3D render of. I need to do it in a vector drawing programme and need to know how to derive the angles and lengths of the sides.

    The form is that of the three intersecting golden rectangles rotated in such a way as to produce the appearance of three intersecting planes that fit within a 3 Dimensional Icosahedron.

    I reproduce the figures I am working from here:




    In tracing the rasterised vector artwork above I find that the sides of the parallelograms are about 6.5 and 10.5 cm. When I use trig to find the top right apex interior angle it is 59.4° - or I assume 60° due to the perspective method used...

    Is it the case that the matter of creating a parallelogram out of a golden rectangle does not in fact interfere with the properties of the object?

    I figure that all I need to work out is how to take a golden rectancle and shear it in such a way that it is still a fibonacci shape...

    I am working off the basis of a golden rectangle in the simplest way I can:

    10cm x 16.18cm

    Is this possible or is the very nature of shearing the golden rectangle to create the illusion of 3 Dimensions going to destroy the phi relationships - and if so can I just pick a suitable interior angle and then rotate three copies of the created parallelogram in such a way as to get the effect...

    I will be then using colour and transparency blending to create a geometric logo/design element in stereo...

    Many thanks in advance to anybody who can provide any (ph)insight
    Regards,
    Will

    [update]
    these are the constructs I have managed this afternoon using only very elementary assumptions about the geometries involved...

    [/update]
    Last edited by stardotstar; June 10th 2008 at 11:19 PM. Reason: update
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  2. #2
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    Joined
    May 2008
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    Solved

    I have found a solution to this - the construction of a golden parallelogram is from the pentogram enclosed in a pentagon.

    It has the special property of the shorter diagonal being equal to the longer parappel sides and the ratio of the short to long sides being Phi.

    I am continuing to pursue this design and will post more if I come up with anything of note.

    \\'
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