Tikz movies (?) of a rotating cube

 From fewer than 90 lines of Tikz code . . . 

 

. . . I can create a 100-page pdf that gives me a flip book of a rotating necker cube.

Here's a geogebra applet that allows me to rotate a cube in perspective, with a "large" viewing distance (relative to the distance to the cube).

This looks fairly reasonable to us, even though we're not at the right viewing distance.  

But if we move the image of the cube far from the viewing target (or equivalently, decrease the viewing distance), the cube looks like a dumpster when were are not at the right viewing location.  

Similarly, if we rotate the cube, it now looks like in changes shape as it rotates (again, when we are not at the right viewing location). 

Question:  How to code this in tikz, to make it into a flip book?

Anamorphic vault (Bramante's Santa Maria presso San Satiro)

Anamorphic art creates "useful lies", even in places of great truth.  Perhaps "lies" is the wrong word; a more positive interpretation is that these illusions can create metaphors of worlds beyond the world we can see and touch.  The example below is just one of many.  The words that follow are a reposting of a small snippet of a longer (and beautiful) post by my colleague, a professor emeritus of Art History at Franklin & Marshall, Tyko Kihlstedt.  

Santa Maria presso San Satiro

Donato Bramante may have had little to do with the basic design of this church, but his training in the new science of illusionistic perspective painting was essential in order to complete Santa Maria presso San Satiro as a properly-functioning church. This is because there is only another three feet of actual space behind the priest whom we see standing at a lectern on the left. The entire choir with its arching barrel vault and receding walls, defined by three arches & pilasters, is not physically real. This choir and all of its architectural elements was painted by Bramante to provide the illusion of real spatial depth.

Nave, Santa Maria presso San Satiro, 1472-1482, Giovanni Antonio Amadeo and Donato Bramante (architects), Milan, Italy

There was, you see, no room for a choir. If you scroll up to the first photograph of San Satiro, you can see the dome and lantern of Santa Maria presso San Satiro poking up behind the older chapel. Where the choir should have been, on our far left, we instead see a road slicing through. A real choir would have needed to occupy the space of this road. The road remains. Bramante simply ‘replaced’ it by painting a powerful Renaissance barrel vault on a flat wall, so creating a major tour-de-force of  illusionistic perspective.

Choir, Santa Maria presso San Satiro, ca.1482 ff, Donato Bramante, Milan, Italy

By the way, those wonderfully expressive terracotta heads in the circular frieze of San Satiro could well be designed by Bramante. They are, most definitely, later 15th-century pieces of sculpture from the hand of a very good artist.

Student Art gallery

 We had a great semester in my Perspective and Projective Geometry class.  Here, by popular request, is an art gallery featuring selections from our final projects.  I'm just delighted with how well these came out!

Winding down a crooked street . . . 

. . . and arriving back home!

Keyli Motino has an homage to a beloved building near campus.

Ming Gao says we had fun in MAT245!

Riley Cox goes green with this greenhouse.

Sarah Peichel did some really good work figuring
out how to get those roofs to meet up correctly!

"Breathtaking Vertigo" with an unusual perspective
from Tech Oh.

Wendy Huang's room reminds me a bit
of M.C. Escher, one of my favorite artists.

This class really gave everyone's brain a workout,
so it's good to include the gym!

Mark Brugger has a cloud
(drawn per class regulations with a straightedge)
above the house.

These students made my last semester of teaching a really wonderful one for me, and I hope they (and their classmates) stay in touch!

When the sun paints a picture

No matter how many different times I see people doing window taping activities, I can always be surprised and delighted by what occurs.  The pair of pictures that follows was one of these "lightbulb moments" for me.




Rather, I should say it was one of those "rising sun moments".  Gordon Williams of University of Alaska Fairbanks sent me these images, together with the following message.

Hi
Don’t know if this has happened to you before… but my students’ window drawings resulted in an amusing example of a projective transformation when the sun came up this morning! (images attached - I think the first pair is best!)
Enjoy!

Safety Art

Whatever uses people may or may not find for mathematics, it's reassuring to know that at least art is so useful that it can save lives.  (And it can entertain at the same time!) 

Two of my former students separately emailed me about this awesome street intersection:


from Brittany Saunders

I hope this email finds you well! Just came across this article on LinkedIn and I thought you'd be interested since it really is projective geometry being put to use in real life :)
https://www.boredpanda.com/3d-pedestrian-crossing-island/

from Rob Burkhead

This made me think of you and F&M today. Hope all is well. 

https://www.boredpanda.com/3d-pedestrian-crossing-island/
Cheers,
Rob 

I love that the added shadows give the impression that the crosswalk planks are floating.   The link that my students goes to an article titled "Town in Iceland Paints 3D Zebra Crosswalk To Slow Down Speeding Cars".   Check it out!

Different perspectives: literal and metaphorical

Professor Julian Fleron of Westfield State College recently shared an awesome video his students created, illustrating (literally) that different perspectives can be both literal and metaphorical. 

Fleron had attended one of our Viewpoints workshops at the turn of the millennium, and has been using some of our materials in his incredibly innovative classes.   He and his colleagues run a nationally-known project called "Discovering the Art of Mathematics", and he described his student project in this blog post. 

Perspective by the numbers (with Excel)

If you've ever wondered what it might be like to draw perspective pictures using coordinates and linear algebra, this video might be the one for you. 

This 21-minute video was made by mathematician Ray Beaulieu from Sul Ross State University (but now working for the government), as a follow-up to a summer Viewpoints workshop.  He wanted to let other workshop participants know what he'd done during the rest of the summer . . . and what he'd done knocked our socks off.

Using Excel to draw (and rotate) 3-d objects in perspective