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INSTRUCTIONS FOR PERSPECTIVE DRAWING OF

TILE FLOOR USING GEOMETER’S SKETCHPAD

1. Place and label point *A* near bottom of screen.

2. Place and label point *C* along horizontal line also near bottom of screen, several inches away from *A*.

3. Construct line segment *AC*.

4. Construct midpoint of segment *AC*.

5. Label the midpoint *B*.

6. Construct the midpoint of segment *AB* and label it *D*. To do this you will have to select points *A* and *B* and construct the segment *AB*. This will appear on top of the segment already constructed. Then you can select that segment and construct its midpoint.

7. Construct the midpoint of segment *BC* (as in step 6) and label it *E*.

8. Place and label point *Z* several inches above *AC* and approximately over the point *B*.

9. Construct segments *AZ*, *BZ*, *CZ*, *DZ*, and *EZ*.

10. Construct a line through *Z* parallel to *AC.*

11. On new line constructed in step 10 above construct and label point *X* to the left and several inches away.

12. Construct *DX*.

13. Construct the point of intersection of *AZ* and *DX* and label the point *L* .

14. Construct *BX*.

15. Construct the point of intersection of *AZ* and *BX* and label the point *M.*

16. Construct *EX*.

17. Construct the point of intersection of *AZ* and *EX* and label the point *N* .

18. Construct ray through *L* and parallel to *AC*. Note that this line goes through the intersection point of *DZ* and *BX*. Construct that intersection point and label it *P* .

19. Construct ray through *M* parallel to *AC*. Note that this line goes through the intersection point of *EX* and *DZ*. Construct that intersection point and label it *Q* .

20. Construct ray through *N* and parallel to *AC* .

21. (Optional: You may “clean up” the drawing or skip this step.) Select and undo the construction lines *DL*, *MB* and *EN*. Color in “tiles” to make drawing appear more like a tile floor, by constructing the polygon interior and selecting a color from the display menu.

22. Optional: You may want to construct columns at the edge of your tile floor. To do this, construct a line segment at *A* perpendicular to *AC*. Make this line segment a bit longer than *AD*. Label the upper endpoint of this line segment *S* and construct *SZ.* Then construct line segments from *L*, *M*, *N* each perpendicular to *AC* and ending on the line segment *SZ*. These constructions will take some thought, because you will probably have to construct certain lines and later delete certain parts of the line. Think carefully about how to do this.

23. a) Label the vanishing point.

b) Label the horizon line.

c) Measure the length *XZ*. This is the viewing distance of the painting.

d) Measure the line segments to find the ratios of *PL* to *AD* and of

*XZ* to (*XZ+AD*)

24. The Renaissance artist/mathematian Piero della Francesca (1420-1492) said that “the ratio of the first line to the second line is always proportional to the ratio of the distances from the eye to the first line and from the second to the eye.” Copy and paste this quotation onto your drawing and measure the following lengths to show that these ratios are the same: *MQ* to *AD*; *XZ* to (*XZ*+2*AD*).

25. If you have constructed the columns in step 22, find the ratio of the length of the second column to the first (*AS*). Show that this ratio is the same as that of *PL* to *AD*. Similarly, find the ratio of the length of the third column to the first and show that it is the same as that of *MQ* to *AD*.

Rick Faloon (The Ross School), "Teaching Leonardo: An Integrated Approach - Tiling a Floor Using Geometer's Sketchpad," *Convergence* (June 2010), DOI:10.4169/loci002172