Projective Geometry Lesson 3

Hexagons to the Horizon

Welcome to Lesson 3 in Projective Geometry, on a potentially idiosyncratic course charted by me alone!  I’ve been teaching a great bunch of 11th graders for the last two days, and I taught them the very same drawing that I taught you last time. We will pick up from that drawing, so I hope you have made one and you have it handy. You’ll also need three contrasting colors of crayon or colored pencil to join in today.

I want to urge you one more time to actually do the drawings along with me.  There is a small learning curve, plus the hurdle of just getting some paper, a pencil and a ruler or other straightedge together.  But, it has such great benefits for your understanding to participate with your hand and eye as well as your mind.  If you just read along, that is also fine, but the difference is a real one. It can also be a very pleasing meditative drawing exercise in and of itself, rather like braiding or crocheting.

We are drawing a Projective Net with three equidistant points on the horizon line.  That first horizontal line is actually serving as a kind of horizon, isn’t it? The word “horizontal”, meaning “perfectly side-to-side” comes from the experience of the visual horizon that we see when looking out over an expansive landscape (like out over an ocean or Lake Michigan).  I mentioned last time that when you complete this drawing, your eye and imagination will naturally collaborate to make this drawing appear to have depth.  The triangles are “coming out at you,” as you move down the paper (and flowing right off the page), and they are receding into the distance of the horizon as you move up the page.

There is a coloring exercise that you can do to enhance that “3-D” quality that I’d like to share with you now.  Get three colors of crayon or colored pencil.  Then find within the pattern one hexagon (six-sided figure) and color it in.  A hexagon would consist of six of the triangles nested together:

Now find another hexagon that shares a side with the first one you colored, and color that one with another color:

Finally, identify a third hexagon that shares a side with both of the previous two, and color it in a third color:

You should now be set up to color lots of other hexagons that you can find by choosing the color that is missing in any open space adjacent to what you’ve already colored.  Here is a rather hastily finished coloring job on my drawing.

With this patterned coloring, the drawing really starts to look like a tiled floor stretching all the way toward the distant (how distant??) horizon, and stretching “beyond the page” in every direction.  The actual drawing process has some limitations, but I hope you can see that if you were able to keep drawing in your imagination, you’d fill the page with hexagons, all the way to the horizon. 

There are several wondrous and mysterious things about this drawing that I’d like to highlight here.

1. All you have to do to make this drawing is place three equidistant points on a horizon line, draw one line from the middle point, and two lines from the left point, and the rest of the drawing is automatic.  All the new intersections line up of their own accord and point back to one of the three points.  Why does that happen??  I hope you’re curious.

2. There is something else hidden in this Net that I’d also like to show you now.  If you lay your straight edge or ruler horizontally on your paper, you will find that the intersections also line up on parallel horizontal lines.  Here’s another Net I made that shows that (I drew in two of those parallel lines):

Can you find those horizontal lines on your drawing? How or why does that happen?  We will investigate this further in the next installment.

I hope you enjoy making this drawing.  It can be quite meditative and beautiful.  I’m intentionally sharing rather rough looking drawings with you so no one gets intimidated, and you all will try it out for yourselves! 

Another note that should be obvious: you don’t have to color in hexagons. You can choose to color in different combinations of triangles to make different tiling patterns. We’ll explore that soon, too.

Send me your likes, puzzlements, reactions and thoughts please . . . And any pics of cool drawings that you make!