See SeventhCities for finished product

Renaissance and Pre-Geometry Perspective
This is an integrated unit on the Renaissance in Social Studies, Geometry in math, and Photoshop on Computer. I will speak first on the Perspective Drawing, Art, Geometry and Computer part of the unit. I am going to explain how my students created these drawings in order to explain how we teach them to think about the Renaissance. I teach perspective drawing and the computer project in order to give boys a Renaissance experience while at the same time preparing them for geometry. As perspective art in the Renaissance led to Toscanelli’s perspective map of the earth which Columbus persuasively used to sail west, our perspective geometry project leads to creative art done while thinking, reading and writing about Renaissance history.

It all begins with the horizon line. When we gaze out to sea and ocean meets sky, the horizon just is. When we direct our gaze to a vanishing point, that is an aesthetic as well as practical act. When we turn our heads to create two vanishing points at each end of the horizon, to differentiate foreground from background, that is a mathematical as well as aesthetic act. The horizon line and vanishing point placement in a drawing create the point of view, an echo of the Renaissance humanist belief that man “is the measure of all things” from art to maps of a round earth drawn on flat paper.

Our project begins with rectangular solid drawn in one-point perspective:

Overhead #1.

Notice how vanishing lines determine the sides of the solid; the
back vertical edge is artist’s choice.

Overhead #2.

The box above, and the box below the line of horizon.

Overhead #3

Next, students draw a cube below the horizon line, in two-point perspective. Now lines to the vanishing points create the angles at which the sides and back vertical edges are drawn. The vertical edges are still drawn vertically on paper. Lines parallel to the horizon remain parallel.
The vertical front edge, horizon line and placement of the vanishing points are aesthetic decisions. But from there angles of the sides belong to the geometry of mathematics. The point of view demands the mathematical precision of the draftsman.
If an object, building or city skyline is to be drawn so close to the viewer that part is below the horizon and part is above, some city scape axioms, or rules are required to keep the drawing in perspective.
Just as Isaac Asimov had three laws for robotics in “I, Robot,” here are Mr.McSpadden’s 3 Laws for drawing cityscapes.

Overhead for Rules
1. There is one horizon line with two vanishing points.

2. All vertical lines remain vertical. Lines perpendicular to the horizon line remain perpendicular.

3. All other lines are drawn to one or the other vanishing points, with two notable exceptions:
a) Any naturally curved line, such as the curve of a sphere is foreshortened, and
b) Any naturally angled-line such as the side edge of a pyramid is drawn with a different angle in perspective drawing.

There it is: a closed system with only three postulates, from which all lines may be derived. Notice, “derived,” not “solved.” Only three rules and yet the devil lies in those details. Our correct method for solving the problems posed by # 3 is trial-and-error: what “looks right” by comparing the drawing to the skyline of SF outside our window. Thus, an intuitive aesthetic judgment is used to induce a mathematical system. “Leonardo da Vinci once began one of his writings on painting by saying,” Let no one who is not a mathematician read my works.’” (M. Serra, Discovering Geometry)
Since our school is on a San Francisco hill with views of the Golden Gate Bridge, bay, and city, my students are quite familiar with a city skyline. What “looks right” on paper is easily held up the window for real perspective views. What precede all this is some science lessons on the eye, and how we see. Throughout this process, the question is posed to the boys: “How do you draw a true line on paper?”

Overhead #4 Kentaro

Overhead#5 Luke

This project is the precursor, the set of experiences that prepare the boys for the five postulates of Euclid and deductive reasoning of geometric proofs. The “Common Notions” of Euclid are taught in beginning algebra. One example of Euclid’s common notion is the reflexive property: two things both equal to a third must equal each other. From the postulates, “assumed to be true without proof”, the basic constructions of line and angle bisectors, perpendicular bisectors, construction of parallel lines, and so on, theorems are systematically developed through logical proofs that then lead to further theorems. Such a mathematical system demands some prior experience and prerequisite knowledge. This project of a true line drawn with 3 simple rules is a simpliflied structure of the rigorous logic of formal geometry.

The 12-year olds who while studying the Renaissance make their own perspective drawings will continue with pre-algebra. As 13-year olds they will study beginning algebra through the quadratic equation.
Concurrently, half of the 8th grade class studies hands-on, inductive geometry, using Michael Serra’s Discovering Geometry (Key Curriculum Press, Berkeley). Serra uses a transparent square of paper that he calls patty-paper those folds as origami paper. Patty-paper begins with its own postulates borrowed from Euclid and expanded:
Any two figures on separate patty-papers that fit exactly on top of one another are declared to be congruent without proof.
All students begin with a common but different given, such as 30 differently sized and shaped acute scalene triangles, proceed through steps, and then make a generalization based on common results with no contradiction. These demonstrations of theorems are called conjectures until formally proved (Euclid). The boys then go to high schools that place then into year of either inductive or deductive geometry, depending on placement tests.
It was Alberti, who wrote the book on how to paint in Renaissance perspective, who gave us the rectangular grid to aid the foreshortening of perspective. The ideas were certainly used by many renaissance painters, to say nothing of Toscanelli and his map, which Columbus carried to the new world. I must give credit to James Burke, for his film, “Point of View” in the excellent series, “The Day the Universe Changed”. His film made the first connection that I was aware of between Alberti, Renaissance painter and writer, Toscanelli, supreme problem-solver, Brunelleschi, architect of the dome in Florence, and Columbus. On gridded paper or graph paper, drawing horizontal lines to a vanishing point focuses the boys’ attention on perspective. It is the drawing of a city, line after line, window after window, building after building, which reinforces the necessity of a closed system of rules which breaks up the grid into the perspective of distance as seen by the eye that creates the readiness for geometry. This discovery method is one way of preparing boys for the logic of postulates that lead to theorems.

FOR GEOMETRY TEACHERS
As a young naïve banker and teacher I was puzzled by people who didn’t “get” the logical system of geometry or algebra. Memorize the postulates; understand them and their constructions and you could puzzle out the proofs. Using some guess and check, logical deduction and construction you had a reasonable chance of working out the proof. What I realized later was that in trying to oversimplify algebra or geometry into rote repetition rules, for many people any new problem was a “gotcha”. It was difficult to teach that hard thinking had to precede the practice that led to mastery. For many people mindless repeats of the arithmetic, math, or algebra algorithm didn’t lead to successful learning or understanding. For some people, no matter how many trees they examined, they could never conceptually leap to the abstract idea of forest. In Euclid’s geometry one starts with forest and constructs the meaning tree by tree. This project is a scale model of a forest: learn three simple rules and apply them over and over until system mastery results. Euclid’s achievement is often unrecognized or neglected. His formal deductive system was a great accomplishment of western civilization. Even today, the best geometry texts and teaching methods are those that stay truest to Euclid.
The pedagogic problem is that Euclid demands a high level of readiness, both in mathematics and in intellectual rigor. Most modern geometry programs teach the readiness and small bits of Euclid, and ignore the intellectual, logical mindset that Euclid and geometry demand.
Michael Serra’s Discovering Geometry uses an inductive approach for the first semester, giving proofs by construction demonstrations, and then moving students into Euclid’s logic. It also has exercises is deductive logic to prepare students for proofs.
Trial-and-error, guess and check, are both ways of moving the furniture around in the den. Understanding spatial relationships and how 3-d space relates to 2-d paper drawings is a better way, especially if you happen to be the one moving the couch.
This problem forces boys to learn the system (postulates) first; then apply them over and over. You can draw a city skyline using only verticals and lines drawn to the vanishing points, (rules one and two) but it has all the excitement of the skyline of Peoria, Illinois. By exploring the exceptions, rule 3, the boys are led to ponder, “How do you make new rules (theorems) in a closed system?” It is this thinking that prepares boys for geometry.

Mathematics is probably the worst of all subjects for ignoring its own world history. Even the best math texts present problems as if they were just created the night before, with no context or meaning. This is an attempt to create a math problem, perspective drawing, following three simple rules, that has a historical context, and was very real at the time. Nowhere in the modern math curriculum is problem-solving given a course of its own. The NCTM definition of problem-solving is “what you do when you don’t even know where to start”. Problems in real life are often like that. I don’t mean math courses where the odd-numbered “problems” are answered in the back of the book. Such exercises are necessary, but not problem-solving. Problem-solving means using mathematics to create further meaning in the context of a real world situation. Creating your own city-view in two-point perspective is a Renaissance and a modern problem, something city planners have not achieved either. Problem-solving demands bringing all knowledge to the effort, and what a better model for that than Renaissance men who acted as if art, science, and mathematics, architecture, map drawing were all one subject to be tackled. Problem-solving demands some life experience to be brought to bear on the task. But how do you gain the necessary patience, endure the frustration, acquire persistence or determination? All too often I have dealt with upper middle-class parents who insisted that their sons be held on track to become executive problem-solvers—but along the way insist that their own sons not encounter any real- life problems. Where do you get the practical knowledge that may give you clues?
I have a student in my advisory that is both delightful and imaginative- yet he is programmed from morning until night. After school he has fencing practice, saxophone lessons, and language lessons. Throw in soccer practices and games, drama in spring, and he is busy until 7pm nightly and all day Saturdays. With the academic load of Stuart Hall, he is often busy nightly until 10pm. Yet his scheduled life is all too typical of most Stuart Hall boys. In my Friday morning advisory period, he and a friend are allowed 15 minutes to “play” a game of their choice. It’s his only weekly free time.
Some where in my childhood between electric trains, building models, reading maps at boy scout camp, and a student job in a kitchen cabinet shop, I acquired the prior knowledge or experience of angles, measurement, and similarity, that made geometry proofs seem logical and easy to progress through. What kind of “play” leads to Euclid? Where do modern boys, in their programmed lives, whose amusements are video games, discover knowledge of right angles? What experiences are they having that will make Euclid achievable without instant amnesia after the final?
When my students have successfully completed a line drawing of a city skyline in two-point perspective, with me as final arbiter of “correctness”,

Illustration B&W city line drawing

they then scan it into the Mac computers using “PhotoShop” software. It is at this point the Renaissance project and major conceit begins: Students are to color their cities, and then change them completely. As Renaissance artists changed from the medieval God’s eye view of the world to a humanist one, so are modern boys to use the computer to change their universe. The challenge is, “If you could see the world a different way, how would you see it?” Modern boys have the advantage of 400 years of art history and styles to emulate, and the break-up that is at the core of much modern art. The only other stipulation is that no two renderings can be alike.
Boys now get a further insight into the nature of real problem-solving. One often has to solve a series of sub-problems to discover the real nature of the real problem. Now the boys have a new problem: they have had introductory classes on Photoshop, but never at this deep level. Now they have to master a new logical system to begin to discover how they will change their world. PhotoShop rules are a good metaphor of how the system of geometry allows you to see interior design, architecture, city planning, and common forms in a different, new way. Good interior designers, good architects, good contractors have an “insight” into space. They can see flow, focus, harmony of proportions—which is simply geometry applied.

In creating new city skylines, boys scan in their own black-and-white cityscapes and they create a new background layer which they then color. Each new color may be a new layer; each effect or change requires a new layer created and saved. To get to the finished products you will see shortly may require 20 or more layers. Also, as layers are saved, boys begin to eye the results critically. To return to a previous state, boys simply delete the last layer. Any art that results from their efforts, even if accidentally achieved, requires severe editing of the layers of unsuccessful effects.
PhotoShop layers, saved and piled on top of each other, represent another logically closed system. It proceeds from givens, definitions, but allows infinite possibilities. No matter how imaginatively PhotoShop commands are applied, they are saved and preserved by a structured language logically applied. The mathematics behind the art is deeply buried, but clearly there. But once students can control some possibilities in PhotoShop, of color, form, and texture, there is still the problem of changing the universe into many versions, and than aesthetically choosing one for final submission. Layers upon layers of possibilities begin to pile up as boys spend hours in the computer lab. Even when a boy chooses an abstract manipulation as his best art, it began as a logical construct scanned as a postulate into a system—a core of geometry. I make every boy do his project layer by layer so that I can see its history to help him gain the “insight “of geometry. Why study the Renaissance during history to do a geometry project on computer?
Imagine the astonishment of a Lorenzo Medici being told after he was stabbed in the Cathedral during Mass that he was living in a “rebirth”. Imagine the astonishment of a Cellini, fighting duels, suffering agony from a sexually-transmitted disease, arguing with customers and patrons alike, that he was living in a time famous in history. There is not much in the Autobiography of Benevenuto Cellini that one can offer a 12 –year old to emulate, yet his art remains, a master of molds. What was it to be a Renaissance man or woman and yet never know what exactly was special about the age, except perhaps that some were living a life as good as the ancient Romans, ”a good, human life”.
Yet what was the Renaissance? Constant war between city-states, assassinations, economic intrigue and warfare, sexually transmitted plagues without cures, scandals and corruption
In the Church, and yet that sounds exactly like today’s news.
In Orson Welles’ “The Third Man,” the character Harry Lime remarks that Switzerland had 400 years of peace and produced only cheese and cuckoo clocks, while Italy, having political, social, religious and economic upheaval, gave us the Renaissance.
It wasn’t until 200 years later that the term “renaissance” was applied to the period and the city of Florence. Two hundred years from now, how will historians view Silicon Valley and the personal computer with Internet connection? If you were living in a Golden Age, how would you know? If you were the late SF newspaper columnist Herb Caen, would you be surprised to keep meeting people who insisted that “now” was the Golden Decade of San Francisco—and that happened to Herb Caen for over 60 years?
The conceit is that boys were using computers to emulate the humanist spirit of the Renaissance: seeing the world in a different way. To extend the conceit, perhaps someday the personal computer will be viewed as the beginning of some period of history YET TO BE NAMED. Yet many modern boys are as overworked as any apprentice boys of the 1400s. The spirit of creativity seems more akin to play than work, although great solutions or inventions usually require copious amounts of both. Mathematicians, scientists, and artists all “play” with their problems. So view this slideshow not just as pre-geometry or as neo-renaissance project art, but as 12 year olds in directed play—a “play” that must precede actual learning.
Example of student writing for Renaissance Project