The Alhambra

1. By Kara, Valerie, Sam and Angela The Alhambra

2. http://www.youtube.com/watch?v=-3DjvxU0EQs Intro to Alhambra video

3. Islam in Spain: Al-Andalus CONQUEST and UMAYYAD RULE • Granada was once part of empire conquered by Muslims aka “Moors” • Conquered 711 • Series of shifts in power over next few centuries • Major center of knowledge and education

4. Islam in Spain: Al-Andalus FRAGMENTATION and DECLINE • Broken into series of semi-independent states circa 1031 • Civil war, battle with Christian kingdoms • Granada surrendered to Spain in 1492

5. Completed by Yusuf I and Muhammed V Ibn Nasr and the Nasrid Empire The Reconquesta and the Alhambra Decree Charles V's additions http://www.saudiaramcoworld.com/issue/200604/# The Red Palace

6. A wallpaper pattern is one that has translation symmetry in two directions (such as left/right and up/down). A frieze, or strip, pattern is one that has translation symmetry in one direction. A rosette pattern is one that has no translation symmetry, just reflection and/or rotation symmetry. http://britton.disted.camosun.bc.ca/alhambra/index.html Tiling/Symmetry/Patterns

7. Wallpaper pattern

8. Rosette Pattern

9. Frieze Pattern http://www.ted.com/talks/lang/eng/marcus_du_sautoy_symmetry_reality_s_riddle.html

10. The Golden Ratio, a common misconception • Complex geometry derived from Pythagoras • Rectangular proportions • Present throughout the architecture • Discovered by Professor Antonio Fernandez-Puertas • Now a demonstration! Math in the Alhambra

11. Rectangular Proportions 1 1 1

12. Rectangular Proportions in the Alhambra

13. Thanks For Listening! • HW: • Begin with a 1 x 1 square, and verify the proportional rectangles for your self, using the Pythagorean Theorem. • Using Geometer’s Sketch Pad create your own design with a Rosette pattern.

14. HW Continued • Begin with a right triangle with a base and height of 1 (Figure 1). • Select the hypotenuse, and the top point; create a perpendicular line (Figure 2). • Create a circle with a radius of 1 (Figure 3). • Create a line segment from the center of the circle along the perpendicular line (Figure 4). • Connect the end of this segment to the original triangle as seen in Figure 5. • Hide the circle and perpendicular line, and continue this process to create a spiral of square roots (Figure 6). (Be sure to label the length of each hypotenuse.)

15. HW Continued • Copy the image at the right. • Label the center point, at which you could rotate the design 180 degrees and have the same image (with the exception of color). • Label the center point, at which you could rotate the design 120 degrees and have the same image (with the exception of color). • Label the center point, at which you could rotate the design 60 degrees and have the same image (with the exception of color).