ASSIGNMENT # 2

1. ASSIGNMENT # 2 LINEAR RELATIONS & LINEAR PERSPECTIVE DRAWING

3. HISTORICAL ERA: RENAISSANCE ITALY CHARACTERS: LEONARDO DA VINCI & RENE DESCARTES (FR.) MATH/ART FOCUS: LINEAR RELATIONS & LINEAR PERSPECTIVE DRAWING MATH STRAND: ANALYTIC GEOMETRY

8. LINEAR RELATIONS & LINEAR PERSPECTIVE • In mathematics, a linear relation is defined as a relation between two variables that appears as a straight line when graphed on a coordinate system. Much of the grade nine course of study is based on the detailed analysis of the straight line in two dimensions: sketching lines on the Cartesian plane; algebraic representations; key features such as slope, intercepts, and distance; and applications in problem solving (e.g. rate, cost, comparison, etc.). • Linear perspective was a quasi-mathematical system of representing a three-dimensional world on a two-dimensional surface. It was perfected as a visual arts “science” during the Renaissance, first in Italy and then throughout continental Europe. • This second assignment brings together the ideas of both 17th century Analytic Geometry and 16th century Linear Perspective. It fancifully supposes that the Italian artist/polymath, Leonardo da Vinci, and the French mathematician/philosopher, Rene Descartes, could somehow have met each other, discussed their respective sciences, and collaborated on a project.

9. A LINEAR “SMARTWORK” (SECONDARY MATH & ART WORK) • Suppose Italian Leonardo da Vinci (1452-1519) could somehow have crossed paths with French mathematician René Descartes (1596-1650). Suppose further that they had decided to collaborate on a series of special drawings that incorporated principles from both Renaissance linear perspective and 17th century analytic geometry. One wonders what kind of interesting results may have occurred & just how the two “sciences” would have related. • On the large sheet of graph paper provided, create a Renaissance linear SMARTWORK that encompasses all of the following items & conditions: • Cartesian Plane (x and y axes) must be clearly labeled; not necessarily in center • Rendered solids must include at least 1 cube and 1 rectangular prism • Key coordinates (vanishing point(s) & vertices of solids) must be clearly labeled • Construction lines (at least 4) must be labeled with corresponding equations • Antiquated appearance: coffee and dirt stains, crumple marks, singed edges • Signature using Leonardo’s backwards (mirror) writing style • The rendered scene may be interior, exterior, or fantasy. You may use either 1-, 2-, or even 3-Point Linear Perspective. It may be monochromatic (Burnt Sienna was quite popular in sketchbooks) or be rendered in colour. Further explorations might include adding an original invention sketch, experimenting with the shading of a sphere or human hand, personal notes or poetry, etc. The final composition will be assessed for the inclusion of the required components above, and also for creativity & complexity.

11. SW 38

12. SW 38 “It was hard to get ideas for this project but eventually I finished. My SMARTWORK consists of a Cartesian plane with different things in the four sections. In the top left there is a golden rectangle drawn in one-point perspective. It has two lines and the golden section labeled. In the bottom left there is a cube drawn in one-point perspective with two lines labeled. One thing that I noticed about the lines that I labeled is that their slopes are reciprocals. In the top right corner I wrote some stuff down about finding the diagonal of a square. The normal way is to use the Pythagorean Theorem but I was told that perimeter over 2.84 works too. I tested them on the paper and it shows that that way works but it isn’t exactly accurate. Then I tried to find another way of doing it but I couldn’t find one. In the bottom right corner I experimented with different ways of drawing with perspective and wrote a quote by Leonardo that I thought was interesting. In the end, I think I learned a lot about math, art & the Renaissance.”

13. SW 39

14. SW 40 “My finished project is supposed to be the basic things inside a sketchbook that belonged to Leonardo. The cover is just Leonardo’s backwards ‘mirror’ writing. The second page is my shapes and the Cartesian Plane. The third page has two more equations of lines. The last page has my signature and a design that I experimented with. I burned it to look old.”

15. “For my SMARTWORK, I have chosen 1-point linear perspective. I made the vanishing point at the front of the presentation, rather than in the middle. For the drawing, I drew the headstock of a Gibson Flying-V guitar with the tuning heads as the cubes, and the sides of the headstock as the rectangular prisms. To give it an antiquated look, I burned the edges with a candle, and stained it with coffee.” SW 41

16. SW 42 “I started my SMARTWORK by constructing the Cartesian Plane. I labeled my x and y axes. I used separate vanishing points for my cube and rectangular prism. I then calculated the slopes and made the equations. I used wrapping paper to ‘dress up’ my shapes. I cheated writing my words backwards by pressing really hard on the back of the paper and then flipping it over and tracing the lines at the window. I spilled tea on it, made a circle stain with the mug, and crumpled and ripped the edges.”

17. SW 43

18. SW 43 “For this SMARTWORK section of the project, I tried to make it as creative as my last one. I placed my Cartesian Plane in the center of my paper. I then drew a cube in the top-left corner and a rectangular prism at the bottom. On the cube I tried to make it look like there was a square hole in it. I had four equations altogether on the sheet - two for each shape. I calculated the slope of the lines using the given slope formula, and then found the equations. I wanted my project to look old and good, so I took my paper over to my stove and aged the paper by placing it over the hot stove until each section of the paper was brown and crisp, but not burned to ashes. We then took a candle and burned some holes in it for a better effect. I enjoyed this a lot.”

19. SW 44

20. SW 44 “This is my SMARTWORK. I have done three cubes because it looked better than just one. I did tow rectangular prisms because I wanted to show it in two different positions. I coloured the faces so that you would be able to see the original shapes. The barn is made up of one rectangular prism and one triangular prism. All of my equations are done and numbered. I signed my name twice - once in writing and once in printing - and I did it all without cheating [mirror writing without tracing through paper from back]. For an antiquated appearance, I put coffee and tea stains on the project and I burned the edges.”

21. SW 45

22. SW 46

23. SW 46 “For my SMARTWORK I decided to draw a picture of a perspective room. The Cartesian plane is located in the middle of the page. The coordinate (0,0) is also the vanishing point in the picture. I drew the rendered solids on the Cartesian plane. The cube is in the bottom right corner and the rectangular prism is in the bottom left corner. The key coordinates (vanishing point & key vertices) are labeled. Four of my construction lines are labeled with corresponding equations. The construction lines that I labeled are the lines from the roof, floor and walls. To make it look antique, I wet a tea bag and spread and dripped it all over. TIP: when using a tea bag to antiquate and using washable markers to colour, be careful because it will smudge and run. I also crumpled it up and let it dry crumpled up. I wrote my name backwards. It wasn’t too hard, but it looks as bad as when I write it frontwards.”

24. SW 47

25. SW 48 “I decided to make a bunch of shapes that appear coming close to you. I chose that because I thought it looked neat. After I was done drawing it, I went outside and burned the edges to give it an older look. I also spilled some orange juice on it. I picked out some lines and found their equations and wrote them on the lines. Finally, I wrote my name using Leonardo’s backwards writing style.”

26. SW 49

27. SW 49 “To make my project look old, I crumpled the edges and smeared dirt on it. For ‘Mona’ and the cubes I only used one vanishing point and for the rectangles I used two vanishing points to make it look like a 3-D appearance. In the right-hand corner I wrote a message on success in Leonardo da Vinci’s backwards writing. The equations for the slopes are right on the paper and the slopes are clearly labeled on the lines. In the bottom left-hand corner there is one of my inventions: a solar-powered boat with attractive designs on it. The Cartesian Plane is also centered in the middle of the page, with the x and y axes clearly labeled. I chose the word ‘Mona’ because it is my middle name and also it is a famous painting. My name is signed backwards in the right-hand corner.”

28. SW 50

29. SW 51

30. SW 51 “I drew a picture of a building and took the vertical lines of each segment of the building and used those to find the slope of each segment by using the given formula. Then I labeled the Cartesian Plane and all the vertices that I had selected, and the vanishing point. Then I did Leonardo’ backward writing style. Next, I put instant coffee in a cookie tray or pan, mixed this with room-temperature water so it would dissolve, stirred with fork, placed paper in cold coffee, let it sit for two minutes. Then I turned it over, let sit for another two minutes, placed on clean dry sheet, placed in pre-heated oven @ 250o F, kept a close eye on it until it started to bubble, took it out, burned the edges, folded it up and brought it to you.”

31. SW 52

32. SW 53

33. SW 53 “For my SMARTWORK I decided to create a 1-point linear perspective drawing using a square, a rectangle and a triangle. I decided to draw lines from these shapes to the vanishing point to show the shading that occurs: it obviously gets darker as it gets closer to the vanishing point, giving the illusion of the shape disappearing into the vanishing point. I decided to find the slope for at least one construction line for each shape. This was then used to create equations for each of the relevant construction lines. These are labeled on my SMARTWORK. To give my work a more interesting creative look I decided to make sketches of hands in various positions and places. I chose hand positions that would fit into the shapes and maybe even blend with the lines of the shapes. I then decided to burn the edges of my work to give it an aged appearance, but it did not brown to my satisfaction. So, I made some coffee and spilled a little around the edges, slopping it in a few places to complete the look.”

34. SW 54

35. SW 54 “I decided to make some sketches of inventions that I came up with, like the ‘Hover Board,’ with a water-powered engine. I also tried to make the sketches realistic, and they almost didn’t make it through the burning of the edges because they are along the outside of the sheet. The other inventions are also neat because I thing that they could actually be made if some one put their mind to it - even if they don’t have any practical uses. What a great idea to just make some crazy inventions, and have fun while you’re inventing. I also had fun burning and turning the paper to a yellowish colour. I think this was a better project than the last one and I enjoyed working on it more. This project also gets you to use and improve your skills with linear relations.”

36. LEARNING IS THE ONLY THING THE MIND NEVER EXHAUSTS, NEVER FEARS, AND NEVER REGRETS. LEONARDO DA VINCI (1452-1519)