Sunday, September 22, 2013

3D Fractal Geometric Drawings: Icosahedral / Dodecahedral Stars

"Dodecahedral Star"
Fractal geometric drawing; arrangement of (3D) Dodecahedra
in the Pythagorean Lute (from subdivisions of a Pentagon/Pentacle)
drawn by hand using classic geometric construction. No computer, pre-existing image or measurements were used to create this drawing.
Compass and straightedge, pencil and ink on archival paper.
Alberto J. Almarza
September 1, 2013

"Icosahedral Star"
Fractal geometric drawing; arrangement of (3D) Icosahedra
(from  subdivisions of a Hexagon; incremental tilling -or fractal tessellation-)
drawn by hand using classic geometric construction. No computer, pre-existing image or measurements were used to create this drawing.
Compass and straightedge, pencil and ink on archival paper.
Alberto J. Almarza
September 18, 2013

Thanks for visiting!
Please feel free to post questions or comments!



  1. how do you get such accurate proportions, lengths and angles in your polygons?

  2. Thanks for asking!
    I use a circular compass and a straightedge. I have learned over years of practice that one can create an isometric view of just about any 3D polyhedron using the proportions embedded in their respective 2D polygonal faces and/or contours: for example, you can draw a cube or an icosahedron starting from a hexagon, or a dodecahedron starting with a pentagon. All the needed angles and ratios are hidden within the basic forms!
    Its the beauty of geometry. :)

  3. Ya, I love geometry too. Could you show me how you draw your icosahedra and dodecahedra? Do you use the golden ratio or other sacred geometry? Do you know some interesting websites or books for information about geometry?


    I really like your drawings by the way.

  4. Thanks!
    I could show you how I do these and other geometric things given that you put the investment in learning it! I teach a couple of classes here in Pittsburgh, and mostly I teach anybody in person who is willing to learn.
    Some of the methods are more simple than others, but they all come from the same basic forms and principles: if you learn the regular polygons and start messing with them, you continue to arrive at greater levels of complexity.
    I did not "use" the golden ratio per se, although it is naturally occurring in many of these shapes, predominantly the pentagon, which can yield a dodecahedron in isometric view. What I mean by that is that, as the 12 faces of a dodecahedron are each pentagonal, then you can draw an isometric dodecahedron by starting with a pentagon, i.e. the one face that is level and directly perpendicular to your line of sight. (In that view, many of the individual line segments are related to one another by golden proportions: they approximate the "mean ratio" or relationship of 1 to 1.61803... the number known as PHI).
    Another sacred geometry form is popularly known as the "seed of life", which is essentially the simple scaffolding for a hexagon created by seven circles. This deceptively simple form contains the information for more complex geometric patterns such as various polyhedra, and it is all embedded in the original ratios and their subdivision, intersection, etc.
    To understand this, start by drawing a hexagon. Then, connect each OTHER vertex to the center, creating three sub-dividing lines. If you look closely, you will see that you have just drawn a little cube. An ISOMETRIC cube. The icosahedron, when looked at "straight" -or at eye-level- can have a perfect hexagonal contour. By this same principle, one can draw an isometric icosahedron starting with a simple hexagon, you see? If you still don't know what I mean, look closely at the two drawings above. You might see the dodecahedra and icosahedra, but you can also spot the basic polygons: the equilateral triangle, pentagon and hexagon.
    I hope this brief explanation can at least get you started into more than just drawing a shape or two, but delving a bit deeper into geometric magic! You will have to put the work in though :)
    There are a lot of books out there, specially nowadays that geometry is having a come back. I have read a bunch and taken a little here and there from each. One of my personal favorites is Albrecht Durer's "Painter's Manual", which may not even be in print, I get it at the library.
    It has Durer's own plates and notes, and specifically a number of compass and straightedge constructions of simple things, like polygons, ovals, spirals and stuff. there is also a myriad of internet content and even video tutorials out there about this stuff. Like I said before, my biggest advice is to start with the basic shapes and see what they reveal after a little bit of unearthing. Geometry is the best geometry teacher, and an ancient one at that!


  5. *picks jaw up off floor*
    Ah, now I've gotta get a straightedge, this is awesome.
    I play with shapes in my sketchbook sometimes, recently I've been trying to figure out if a tetrahedron can be construed from smaller yet mathematically similar ones.
    Unfortunately I'm not very good with 3D in my head, or drawing straight lines freehand, so I haven't figured that out yet lol