THEMOSELYSNOWFLAKESPONGE A giant fractal object discovered by Dr. Jeannine Mosely

HELP CREATE THIS REMARKABLE MODEL FOR THE FIRST TIME IN THE HISTORY OF THE UNIVERSE

A project of THE USC LIBRARIES curated by Discovery Fellow MARGARET WERTHEIM

www.usc.edu/libraries/sponge To find out how to participate, email Tyson Gaskill [email protected]

A USC Community Project at the Intersection of Mathematics, Art, Engineering, and Libraries

In 1926, mathematician Karl Menger discovered the first three-dimensional fractal, which came to be called, in his honor, the Menger Sponge. Ninety years later Dr. Jeannine Mosely, an MIT-trained engineer, built a model of this extraordinary object using 66,000 business cards.

Now Mosely has discovered a relative of Menger’s original fractal—the Mosely Snowflake Sponge. During spring semester 2012, the USC community will construct this unique mathematical form out of 49,000 business cards.

This is the first time in the history of the world, and perhaps the universe, that this wondrous fractal object will be brought into being. We invite students, faculty and all members of the USC community to join with the USC Libraries in this campus-wide celebration of mathematics, art, engineering, and the library as a place of discovery and invention.

The instructions here show how to assemble the basic building blocks of the Mosely Snowflake Sponge. 108 X-modules and 216 Y-modules are required to construct the final object. Instructional videos may be found at www.usc.edu/libraries/sponge.

THE DISCOVERY FELLOWSHIP OF THE USC LIBRARIES THEMOSELYSNOWFLAKESPONGE This event is part of the USC Libraries Discovery Fellowship 2011-2012 programming, curated by Discovery Fellow Margaret Wertheim. CONSTRUCTION GUIDE

Margaret Wertheim is a science writer and curator of exhibitions about the intersection of science, mathematics, and aesthetics. Wertheim is director of the Institute For Figuring in Los Angeles.

Jeannine Mosely is a Boston-based software engineer who specializes in 3-D modeling programs. A pioneer in the field of mathematical paper folding, Mosely is the foremost expert on business card origami and is a leading researcher in the field of curved folding.

More information about the mathematics of the Menger Sponge and the history of Mosely’s Business Card Sponge project can be seen in the Institute For Figuring’s online exhibit: www.theiff.org/oexhibits/menger01.html

www.usc.edu/libraries/sponge HOWTOMAKEANDPANELCARDCUBES MAKINGTHEXMODULE MAKINGTHEYMODULE 108 x-modules are needed 216 y-modules are needed

1. 2. 3. 4. 1. 2. 1. 2.

9.

(6) Add the short rod of 2 cubes. (5) Eight linked cubes. 5. 6. 7. 8.

(6) Add the short rod of 2 cubes. 10. (5) Eight linked cubes. (6) Add the short rod of 2 cubes. (5) Eight linked cubes.

(7) Ten linked cubes. (8) Rotate counterclockwise 90 degrees. 3. 4. 5. 6. 3. 4. 5. 6. Add the last rod of 3 cubes.. 11. 9. 10. 11. 12.

(6) Add the short rod of 2 cubes. (8) Rotate counterclockwise 90 degrees. (5) Eight linked cubes. (7) Ten linked cubes. Add the last rod of 3 cubes.. (8) Rotate counterclockwise 90 degrees. (7) Ten linked cubes. Add the last rod of 3 cubes.. Place one business card across another, centered and at right angles to the first. Fold the flaps down as shown. Separate the 1. 2. 3. (9) Thirteen linked cubes -- two different views. cards. Make 6 for each cube. 4. Assemble the cards so that all the flaps are on the outside, matching short edges to long edges. 5-8. IMPORTANT: You may have noticed that we panel the six faces that look inward toward a void at the center of Adding each of the 6 cards to complete the first cube. Link 2 cubes together by slipping a flap from one cube under the corners the module. Even though they are invisible, we include 9-10. these paneling cards to give extra strength to the model. of 2 flaps on an adjacent cube, then slip the matching flap from the first cube under the other 2 corners of the flaps on the second 12. You must include them! cube. 11-12. Cover the exposed flaps with the facing cards. 7. 8. 7. 8. (8) Rotate counterclockwise 90 degrees. (7) Ten linked cubes. Add the last rod of 3 cubes..(9) Thirteen linked cubes -- two different views. IMPORTANT: You may have noticed that we panel the (9) Thirteen linked cubes -- two different views. six faces that look inward toward a void at the center of the module. Even though they are invisible, we include IMPORTANT: You may have noticed that we panel the these paneling cards to give extra strength to the model. six faces that look inward toward a void at the center of You must include them! the module. Even though they are invisible, we include these paneling cards to give extra strength to the model. You must include them! 15. LINKINGRODSOFCUBES (10) Build a cube one card at a time into this corner. Start by slipping two cards under the flaps as shown. (11) Slide a third card under the flaps shown. (9) Thirteen linked cubes -- two different views. IMPORTANT: You may have noticed that we panel the six faces that look inward toward a void at the center of the module. Even though they are invisible, we include these paneling cards to give extra strength to the model. You must include them! 16.

(10) Build a cube one card at a time into this corner. Start by slipping two cards under the flaps as shown. (11) Slide a third card under the flaps shown. 13. 14. (10) Build a cube one card at a time into this corner. 9. 10. 11. (11) Slide a third card under the flaps shown. Start by slipping two cards under the flaps as shown. (13) Slip the fifth card under the front corners of (12) Slide a fourth card under the first two cards. the same two flaps the third card tucked under, making sure that the front flaps of the first two cards remain outside.

1. 2. 3. 1-2. Panel one face of each of the 2 individual cubes red. 3-6. Make 4 rods of 3 cubes. Apply a red panel card to one face of the center cube in each rod. Note the orientation of the flaps. Two of the paneled faces should have horizontal flaps, the other two should have 17. vertical flaps. 7. Link the 2 single cubes to one of the rods to form a U shape. All 3 of the inner faces of the U should be red. 8. Attach (13) Slip the fifth card under the front corners of the U module to a rod of three cubes as shown. Swing the U and the rod together and slip the marked flaps under the neighbor’s the same two flaps the third card tucked under, (10) Build a cube one card at a time into this corner. (13)(12) Slip Slide the a fourthfifth card card under under the the front first corners two cards. of making sure that the front flaps of the first two the(11) same Slide two aflaps third the card third under card the tucked flaps under,shown. (12) Start Slide by slipping a fourth two card cards under under the the first flaps two as cards. shown. cards(15) Finishremain theoutside. module by paneling the 10 faces that flaps as indicated by the arrows. 9. Eight linked cubes. 10. Add another rod of 3 cubes. 11. Add the fourth rod of 3 cubes. Note that you making sure that the front flaps of the first two form a pair of crosses with RED cards. Panel in cards remain outside. YELLOW six faces in the corners between the crosses, (14) Complete the cube by adding the sixth card and the four faces that are opposite them on the back are paneling the 6 faces that look inward toward a void at the center of the module. Even though they are invisible, we include these to the top, again keeping all flaps on the outside. side of the module. paneling cards to give extra strength to the model. 12. Finish the module by paneling the 5 faces that form a cross shape on one end . A with red cards. Panel in yellow the 4 faces shown below that are not on the ends, and the 4 faces that are opposite them on the back side of the module. Note that you must match the orientation of the flaps. C 18. (15) Finish the module by paneling the 10 faces that form a pair of crosses with RED cards. Panel in (15) Finish the module by paneling the 10 faces that YELLOW six faces in the corners between the crosses, form(13) Slipa pair the of fifth crosses card with under RED the cards.front corners Panel inof the(14) same Complete two flaps the cubethe third by addingcard tucked the sixth under, card and the four faces that are opposite them on the back 19. 20. (12) Slide a fourth card under the first two cards. YELLOWto the top, sixagain faces keeping in the cornersall flaps between on the outside.the crosses, side of the module. (14) Complete the cube by adding the sixth card andmaking the foursure facesthat the that front are oppositeflaps of thethem first on two the back cards remain outside. B to the top, again keeping all flaps on the outside. side of the module. . . D 1-2. Panel one face of each of the 2 individual cubes red. 3-6. Make 3 rods of three cubes and 1 rod of 2 cubes. Apply a red panel card

THEMOSELYSNOWFLAKESPONGE to one face on each rod as shown. Note the orientation of the flaps. Two of the paneled faces should have horizontal flaps, the other A giant fractal object discovered by Dr. Jeannine Mosely

HELP CREATE THIS REMARKABLE MODEL FOR THE FIRST TIME IN THE HISTORY OF THE UNIVERSE 4. 5. 6. 7. A project of 2 should have vertical flaps.7. Link the 2 single cubes to one of the rods to form a U shape. All 3 of the inner faces of the U should be THE USC LIBRARIES curated by Discovery Fellow MARGARET red.Be careful to match the orientation of the flaps.8. Attach the U module to a rod of 3 cubes as shown. Swing the U and the rod WERTHEIM

www.usc.edu/libraries/sponge To find out how to participate, (15) Finish the module by paneling the 10 faces that email Tyson Gaskill [email protected] together and slip the marked flaps under the neighbor’s flaps as indicated by the arrows. 9. One view formof a 8pair linked of crosses with cubes. RED cards. 10. Panel Add in the

YELLOW six faces in the corners between the crosses, (14) Complete the cube by adding the sixth card and the four faces that are opposite them on the back 1-2. Linking 2 cubes. 3. Link more in the same way to form a rod. 4. Link the back half of 2 rods together by slipping flaps from one short rod of 2 cubes. This step can be tricky—please watch the video on ourto the website.top, again keeping 11. all flapsTen on thelinked outside. cubes.side of12. the module.Rotate counterclockwise rod under the flaps of the neighboring rods.5. Pivot the two rods around the back edge to bring the front edges together. Lift flaps 90 degrees. Add the last rod of 3 cubes. 13-14. Thirteen linked cubes showing different views of the . same. 15. Steps 15-18 can be so that you can complete the linking on the front. Tuck flaps in at A and B. 6. Tuck flaps in at C and D. 7. Two linked rods. You can link tricky—please watch the video. Build a cube one card at a time into this corner. Start by slipping 2 cards under the flaps as shown. together rods of any length, and you can link together many adjacent rods to form panels of any size. 16. Slide a third card under the flaps shown.17. Slide a fourth card under the first 2 cards.18. Slip the fifth card under the front corners of the same 2 flaps with the third card tucked under, making sure that the front flaps of the first 2 cards remain outside.19. Complete 12. the cube by adding the sixth card to the top, again keeping all flaps on the outside.20. Finish the module by paneling the 10 faces that form a pair of crosses with red cards. Panel in yellow the 6 faces in the corners between the crosses, and the 4 faces opposite them on the back side of the module.