The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus (2000-1800 BC) and Moscow Papyrus (c. 1890 BC), the Babylonian clay tablets such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Later clay tablets (350-50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.
This is the most dangerous project I have ever done with a group of students. They didn't look like they believed me when I told them that, but there were three very dangerous steps.
#1: Sharp Glass. The students removed glass with sharp edges from old wooden frames of questionable origin. The glass was then carried back and forth across the classroom over several days. After removing the glass from the frame, the glass was cleaned and then covered in clear contact paper. The students then put the glass on top of their reference image, and traced the image with permanent marker onto the contact paper.
#2: X-acto Knives. The contact paper covering each part of the image to be etched had to be removed for the etching cream to work. Students carefully traced the marker lines with x-acto knives and peeled the sections of contact paper away from the glass.
#3: Dangerous Chemicals. Once the design was fully carved out of the contact paper, it was time for the etching cream. We used Armour Etch, which is easy to use, but as their website states, "not intended for use by children." Only two students at a time used the chemical, under close supervision, and never touched it without immediately washing their hands (this was probably overkill; I have used this chemical for years, even touching it extensively, and experienced minimal consequences). The etching cream was left on the glass for at least half an hour, and then scrubbed off using an old toothbrush.
During the rinsing process, the contact paper also got peeled away, and the freshly etched glass was left to dry in the dish rack in the classroom.
After our lengthy discussions of fractals and chaos theory, it was time to focus on fractals in nature. The students each chose a biome to research, and their report must include information on animals, plants, natural cycles, and abiotic factors. Fractals can be found in all of those things. For this art piece, each student chose a natural fractal specific to their biome.
The final art piece for this project is framed etched glass. An eclectic selection of frames was acquired from various thrift stores around Portland, and each student spray-painted their frame the same shade of flat black. I created the example piece below using an image of leaf veins (while my Christmas tree was still decorated).
This time around, we tackled high-level mathematics and abstract art. Not easy for sixth graders. Our masterworks were Number 1A and Lavender Mist by Jackson Pollock, and we studied fractals and chaos theory to understand the natural world and Pollock's brilliance alike.
What exactly is chaos theory, and how does it relate to fractals? This brief documentary created by a kid explains it quite well. The music becomes tedious, so try to ignore those parts, but all of the conceptual stuff is quite well explained. Also, when he starts talking about actual equations, and your eyes start to glaze over, hang in there—the results of the equations are pretty interesting.
So, what does all that have to do with Jackson Pollock? It turns out that the motions that Pollock used when creating his famous splattered paintings are similar to a kicked pendulum. The irregularity introduced by adjusting the length of a pendulum at random (or even regular) intervals results in a fractalized splatter.
"Fractals" weren't a thing anyone knew about when Pollock was painting, so he wasn't doing this intentionally. Art critics also had never heard of fractals at the time, so those who liked Pollock's work couldn't effectively describe why it was so brilliant.
Now look closely. Below is Number 1A (don't be fooled by the title; it wasn't his first).
There are so many interesting parts of this painting; I would love to see it in person someday. The section below is from the center top. It almost looks like a turbulent ocean (which, incidentally, is a natural feature that can be explained by chaos theory and fractals).
Zooming in again and again on the painting reveals that the level of complexity is consistent regardless of how close your viewpoint is to the painting.
Below is Lavender Mist. I love this title. There is nary a drop of lavender paint in this piece, and yet, the overall impression is distinctly lavender. This painting is featured in the movie Mona Lisa Smile.
Again, zooming in reveals continued complexity.
The video below features a pendulum with an "elbow" in it. Though the pendulum is released relatively straight, tiny changes in position become compounded and result in chaotic motion. As the light traces the path, the result starts to look similar to a Pollock piece.
Pollock has an elbow. Also a wrist. Consider the above video with two joints instead of one. Pollock effectively used his arm as a pendulum, but added the "kicked pendulum" motion by utilizing his elbow and wrist joints to create chaotic movement of the brush.
This is not easy to do. Your average person with a brush would not be able to make their arm move in this way. As humans, we try too hard to control motion, even when trying not to.
This is not to say that Pollock lacked control—quite the contrary. Pollock knew exactly where the paint would fall, and he was able to utilize his own joints to make it happen in a way that hasn't been replicated since.
Not a lot of footage of Pollock in action exists, but this little video shows a little bit of what he does with his shoulder, elbow, and wrist to get the paint to fall where he wants in a fractalized way. Also, I love that he has special painting shoes, as you would have to with this style.
Next step for the students: finding fractals in nature.
This project was very personal to each student. In addition to being a portrait of their face, and using words they wrote about their own core values, each project also features the student's own handwriting. The density and disorder of the handwriting varies greatly from project to project, but ultimately all are effective in creating a meaningful and recognizable portrait.
Not every student was completely finished with their projects by the time I started taking photographs in the gallery, but most were. I think only 3-5 were not yet displayed.
I worked with 8th graders for the first time with this art project (many of whom I had worked with as 6th or 7th graders), and it was wonderful to see how they had grown in their art and in their focus. This project involved students identifying an aspect of their own core values, and writing an essay about it. These essays were to follow the "This I Believe" model.
From the This I Believe website: "This I Believe is based on a 1950s radio program of the same name, hosted by acclaimed journalist Edward R. Murrow. Each day, Americans gathered by their radios to hear compelling essays from the likes of Eleanor Roosevelt, Jackie Robinson, Helen Keller, and Harry Truman as well as corporate leaders, cab drivers, scientists, and secretaries—anyone able to distill into a few minutes the guiding principles by which they lived. These essayists’ words brought comfort and inspiration to a country worried about the Cold War, McCarthyism, and racial division."
Modern interpretations of this theme can be found on the This I Believe podcast and are occasionally featured on NPR. Many are tear-inducing, heart-warming, and inspiring—definitely worth a listen.
Upon completion, the pencil lines were erased, portraits carefully trimmed, and then framed.
Les termes qui constituent à notre connaissance une marque déposée ont été désignés comme tels. La présence ou l'absence de cette désignation ne peut toutefois être considérée comme ayant valeur juridique.
En règle générale, la prononciation est donnée entre crochets après chaque entrée. Toutefois, du côté anglais-français et dans le cas des expressions composées de deux ou plusieurs mots non réunis par un trait d'union et faisant l'objet d'une entrée séparée, la prononciation doit être cherchée sous chacun des mots constitutifs de l'expression en question.
The world is not only what we have made it but is also a product of earlier generations' efforts. "Today is the child of yesterday," as the Arabs put it. Thus, to fully grasp the present, we must first understand the past. The great thinkers wrote what they did because they had unusual talents, but they were also strongly influenced by their times. Locke, Madison, Bakunin, Marx, Mill, Hitler, Mao, and all the others can be fully appreciated only in the light of their historical, intellectual, political, social, and economic circumstances. Yet, although these ideologues were influenced by situations particular to their respective ages, they each responded to a common phenomenon—modernization.
The most fundamental feature of this era, the event that has done more to distinguish this period in history from all others and has contributed most heavily to shaping our environment, is the industrial revolution. Industrialization is the latest stage in the chain reaction begun by the scientific method and its application to technology. The shift from making things by hand to mechanized production changed the world dramatically. The reaction to industrialization varied among observers. Some, like Adam Smith, reveled in its potential benefits; others, including Marx, argued that people were robbed of their skills and reduced to being mere tenders of machines, but he looked forward to improved social conditions with the equal distribution of the newly produced bounty; and still more... [something something Hitler].
Text from Political Ideologies: Their Origins and Impact by Leon P. Baradat