The Hexagon and the Construction of Three Dimensional Form

Updated: May 11, 2020

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Today’s post looks into how the eye constructs three dimensional form, using the hexagon. The story is told mostly in pictures. If you find what you see here interesting, please consider reading the companion post The Transparent Box Illusion: Construction of Three Dimensional Form later.



The eye can construct several three-dimensional objects out of this hexagon.


It could be a transparent box leaning to the left:



Or a transparent box leaning to the right:


Or an opaque hexagonal dome viewed from above (this is more obvious if the central lines are blocked).


So the exact same collage could be three different objects.


For these three objects, the incident light is always coming from the same place: above, a little to the left, and in front of the object.




If we put multiple hexagons together, the dome is the dominant image but the other two can be seen. Again, the light is coming from above, to the left, and in front of each hexagon.



Let's add some color:


The domes are easy to see. The individual hexagons are obvious building blocks. The colors separate easily from one another. Now, lets change things around a little bit.


Stare at this at this next design until you begin to settle on one or two faint, evanescent, semi-transparent interlocking three dimensional forms. It helps if you back away from your computer a bit.



Did you notice that this is made up of monochromatic hexagons?


Here are the last two images side by side:

Notice how easy the building blocks are to see in the left hand image. The hexagonal building blocks are three dimensional, and stand alone as important substructures. Not so on the right. The hexagons are not domes and are difficult to see as discreet objects. They do not appear to be important substructures.

What is happening here is a result of the misdirection of incident light. The hexagons are aligned in on the left image, and variably rotated on the right.

If you look let the image on the left, the direction of the incident light is identical for every hexagon. On the right image, this is not the case. If each hexagon is to be an independent three dimensional object, then light would need to be coming from a different direction to illuminate each one. Light does not do that. Light does not shine from one direction on one component and a different direction on another. When the behavior of light and the existence of form come into conflict, the eye rejects form. These cannot be hexagonal pyramids. It is no longer noticeable that they are monochromatic hexagons, because the unit of the hexagon no longer matters. The best the eye can do is create weak, evanescent semi-transparent three dimensional forms. Let’s take this one step further.


Stare at this new design for a little while. Like the previous ones, this is made of individually rotated, identical hexagonal subunits- you can find them if you look carefully.


Below I created the same design using color.





Instead of making the individual hexagons monochromatic, I ignored the borders of the hexagons and chose color patterns to reinforce the geometric forms that I could see in the monochromatic image. Luminosity was not changed for each individual piece.


The luminosity relationships appear random. But you know that they are not. Inside the collage, hexagonal substructures tell a contradictory story about the direction of incident light. The eye is left with no three dimensional truth to find at the level of the hexagonal subunits. So the hexagons disappear from patterns that our eye presents to us.


This demonstrates an incredible power that the eye has- it can specifically ignore contradictory data. You can't control this.

The result: the eye looks for something else. It tells us we are seeing faint, evanescent complex semitransparent geometric structures. That's the best the eye can do with the data given. These faint geometric forms become more and more obvious the longer you look. Partially contradictory data forces the eye to test hypotheses and explore deeper interpretive pathways. Normally faster than a supercomputer, in this case the eye must sort through partially ambiguous information to settle on a solution that ties the form together as well as possible.


Because this takes time, you have the opportunity to observe the eye actively constructing the third dimension. You have literally found your way inside your sensory system, and can watch it work.


If you enjoyed this post, consider reading the companion article "The Transparent Box Illusion: Construction of Three Dimensional Form". If you would like to be on the preview list for upcoming blog posts, please send me an email.


#opticalillusion #illusion #geometry #transparency #collage #visualprocessing #vision #eye


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