The story I tell in my Medium stories, is of an upstart geometry making its way through the sculpture and visual artist domains, with anchoring roots into Ezra Pound era poetry, e.e. cummings for example.
The geometry centers around CHoP, a Concentric Hierarchy of Polyhedrons, and it signs with the branding logo 4D, representing a tetrahedron.
Now let’s tell the story of 2D and 3D, and how the former morphed into the latter through the discovery of perspective in the Renaissance.
3D was the sensational “gotta have it” HDTV of the day, and since the ancient Greeks didn’t have it (they had sculpture) the new upstarts were thinking “hey, we’re the leading edge, not the old fogy ancients”.
The discovery of the “New World” just added to the sense of hubris.
So Aristotle was not infallible after all, what a concept!
“We’ll go with empiricism (experimentation) from now on (scientific method), but with room for a priori Platonic ideas in the mix (logic before history, sense before content).” — That’s a lot of so-called “Western” culture in a nutshell.
The emergence of 3D painting came with a way of doing projective geometry. Painters would pass down these secrets within their studios and guilds, initiating their apprentices, sharing their commissions. Architects followed much the same mold.
The shared goal: to make royalty look good, as the warring families competed for kudos and/or sought to stir fears. The same architect-engineers (the “Leonardo types”, per Operating Manual for Spaceship Earth) would sometimes design weapons, or even torture devices, along with intimidating fortifications.
During the Enlightenment, some notion of universal human rights was established, but took awhile to pan out more globally. Engineers gained “humanity as a whole” as a client only recently.
XYZ it was called, the new way of doing things, and we learn it today.
XY defines the 2D plane of the canvas, and the painter, off canvas, defines the “camera” or “POV” (point of view).
The object of the painter’s attention, call it “the figure”, is likewise in 3D world.
In POV-Ray (povray.org), the positive Z direction is away from the painter’s eye and towards the target. Move in the negative Z direction to come closer, assuming the default camera is on the negative side of the XY canvas.
At the Oregon Curriculum Network website, you’ll find a consistent approach you may wish to adopt for your own courseware: Python or another language is used for Polyhedrons as types, using object oriented thinking.
Once an instance of the Tetrahedron type is defined, in terms of its four corners, using XYZ coordinates, then we have techniques (methods) for computing its volume and surface area.
We also have methods for scaling, rotating and translating the same object. Polyhedrons are mutable.
On the other hand, we may define operations which beget new Polyhedrons from prior ones. Not every operation is a “morph in place”.
These polyhedrons (Tetrahedron, Cube, Octahedron…), with a common Polyhedron ancestor, also gain the means to represent themselves in POV-Ray scene description language. I’ve also written out VRML. The back end is up to you.
Those of you into representing 3D objects in space will have ideas about persisting the data.
Once written out to POV-Ray (.pov files), the polyhedrons are ray traced and show up in perspective. The realism of Renaissance painting is there, along with the ability to create Platonic still lifes. Animation comes later, when still frames get spliced together.
The workflow I’m describing need not be the only onramp into these skills and concepts.
I’m staying on a free and open source path.
The POV-Ray license has its own CompuServ heritage, and grew up independently of the GNU GPL, and we welcome such diversity. No one body of law should be the only law. We need to keep learning from others’ mistakes.
If Aristotle were given the last word on everything, we would never have developed our Z axis as successfully. He was right tetrahedrons fill space (not regular ones): Aristotle was right, remember the MITE (a part of our “CHoP suey”).
At this point, my preferred courseware is likely to spiral through Quadrays as a topic, helping students flex their minds and keep XYZ from becoming the one and only orthodoxy their imaginations might entertain.
Those in engineering may not see the point, since as a practical matter, XYZ controls all the end user machinery. However, our position within the humanities is such that we wish to imbue “4D” with its various meanings, and Quadrays help us remember our CHoP recipes, for generating the CCP for example.
Given Quadrays have for their home base a tetrahedron, helps return us to our poetic roots, in American Transcendentalism especially.