Abcb-Dbcb! by Michel Nylor Western Wshington University The Abcb structure shows up in n mzing vriety of plces. This rticle explores 10 surprising ides which ll shre this pttern, pth tht will tke us through geometry, number systems, rt, music, poetry, higher dimensions, nd more! Did I sy 10 plces? Actully, this one goes to 11. 1. Ruler Tke look t the mrks on ruler in the spce of one inch. There s big mrk dividing the inch in hlf, two shorter mrks dividing those hlves in hlf, nd so on. It s esy to imgine you could keep dividing the ruler gin nd gin infinitely. If you did, you d hve very interesting geometric object clled frctl. 1 2 If the shortest mrks re length 1, the next length 2, nd so on, the pttern of the mrks is 1213121412131215... This pttern shows up in ll kinds of weird plces; let s give it nme.
2. The Nme Insted of using numbers to describe the length of brnches or mrks on the ruler, let s cll the shortest lengths, the next longest b, then c, nd so on. The pttern then becomes: b c b d b c b... or Abcb-Dbcb! This word sounds very much like the mgicin s phrse brcdbr, nd indeed there re seemingly mgicl properties bout this pttern. To understnd the pttern little better nd see how to continue it, let s see how this pttern grows. Strt with n. 1. To grow the pttern, dd the next letter in the lphbet nd then repet everything tht hs gone before (which is just the letter in this cse.) The next step, then, is b. 2. b Continue by dding the next letter, c, nd repeting the b. 3. bcb The fourth step dds the letter d nd repets the pttern: bcbdbcb! The next few steps re shown: 4. bcbdbcb 5. bcbdbcbebcbdbcb 6. bcbdbcbebcbdbcb-fbcbdbcbebcbdbcb It s fun to see how much you cn sy loud. How long would it tke to sy the word ll the wy to z? This word is the bsis for genie nmes in Mggie nd the Abcb Genies (see www.bcbx.com). In the story, Mggie must cll forth genies by sying these nmes, ll the wy to z!
3. The Tree Another wy to represent n Abcb ptterns is with binry brnching tree. In this pttern, one object divides into two, then ech of those divides into two, nd so on, like the brnches on this tree: It is not hrd to imgine tht the brnching nd doubling could be continued infinitely, nd if we could only mgnify our view enough times, we would see the sme ptterns continuing forever nd ever... This sme pttern is followed (in reverse) in ply-off schedule where tems or plyers re pired off with the winner of ech round progressing to the next while the loser is eliminted: A B C D Richie Duncn Mikey Jim Pm Shnnon Dvid Tom Duncn Mikey Pm Tom Duncn Pm Pm If we move from the top to the bottom of the plyoff tree, we ll notice tht the nme closest to the top of the chrt ppers in column A, the next highest nme is in column B, then column A, then C, then bck to A, nd so on. The pttern is bcb-dbcb.
4. Frctls Abcbdbcb is frctl pttern nd it shows up in mny other frctl ptterns. Frctls, like the ruler nd the tree, hve prts tht re similr to the whole if you zoom in nd look closer, you see the ptterns over nd over. Here re three more fmous frctls. The Koch Curve is formed by replcing the center third of line segment with two edges of n equilterl tringle. The Sierpinski Tringle (or Gsket) is formed by removing the center of tringle nd repeting Cntor Dust is mde by successively removing the center third of line segment. Cn you find bcbdbcb in ech of these? Koch Curve Sierpinski Tringle Cntor Dust
5. Number Systems The binry number system is the simplest nd most significnt of ll positionl number systems. In binry, only two digits re used, zeroes nd ones. The vlue of written number is determined by the rrngement of zeros nd ones, with the vlue of ech plce being power of 2 rther thn power of 10 s in our fmilir system. The number 19, for exmple, would be written s follows: 19 (bse 10) = 0 1 0 0 1 1 digits of binry # 32 16 8 4 2 1 vlue of ech digit 2 5 2 4 2 3 2 2 2 1 2 0 power of 2 000000 000001 000010 000011 000100 000101 000110 000111 001000 001001 001010 001011 001100 001101 001110 001111 010000 19 = 2 4 + 2 1 + 2 0 The count from 0 to 8 in binry is: 0, 1, 10, 11, 100, 101, 110, 111, 1000 The bcbdbcb pttern is repeted t infinite levels in this simple counting pttern. The strt of one such pttern is shown to the left. Cn you find others? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 000000 000001 000010 000011 000100 000101 000110 000111 001000 001001 001010 001011 001100 001101 001110 001111 010000 010001 010010 010011 010100 010101 010110 010111 011000 011001 011010 011011 011100 011101 011110 011111 100000
6. Philosophy nd Poetry It s fun to think bout how every decision we mke leds us in new direction, s if our lives re n infinite frctl tree. Here s poem tht reflects those decisions... it hs the structure bcbdbcb. And keep my conscience cler nd bright. I ll do wht I know is right Next time, who knows? I just might! I shouldn t do it, so I thought. I guess I m doomed to live this wy. Now the chnce hs slipped wy Should I do it? Should I not? Mybe on some other dy! It wsn t worth it, I would sy. Now I m full of guilty thoughts But tht s tiny price to py! I went nd did it nywy. Tomorrow I will sty wy. I ll just hope I don t get cught They didn t ctch me yesterdy! Decision Tree by Michel Nylor (College Mth Journl, 32, 3, My 2001.)
7. Puzzles nd Legends A populr puzzle clled The Towers of Hnoi sks plyers to move stck of disks one t time from one of three pegs to nother peg. Only the top disc of tower my be moved, nd one my never plce lrger disc on top of smller disc. The object is to trnsfer the entire tower to different peg in the fewest number of moves. A version of this gme using crds is t the end of this rticle. The key to solving this puzzle lies in the bcb pttern. There is legend of the Temple of Abcbx, where monks move golden disks on dimond spindles. When they hve moved ll 26, the universe will come to n end. Should we worry? The Temple of Abcbx is high top of tower of stone blocks. The blocks re rrnged s such: Drw hlf squre, cut long the digonl to form right isosceles tringle, nd then drw the lrgest squre possible inside of it. Continue by plcing the lrgest squre possible inside of ll the right tringles creted, nd repet. When you decide to stop, you will hve mde stircse of blocks similr to those t the temple of Abcbx. As you climb these stirs, the size of the blocks you step on mkes the bcbdbcb pttern. In the stircse t the ctul temple, this process is crried out 26 times, the centrl block corresponding to the letter z. b c b d b c b At the end of this rticle you ll lso find templte to mke pop-up version of this stircse.
8. Hyperspce Nvigting higher dimensions? Don t get lost! Cll the left-right direction, the up-down direction b, nd the forwrd-bck direction c. Moving will get you from one point to nother you trvel line segment. Moving b will move you round the corners of squre. b Try moving bcb on the cube - did you visit ll of its vertices? c b b Let s dd nother direction, the here-there direction, nd cll it d. To trvel to ll of the vertices of this four dimensionl hypercube, just remember the mgic word: Abcbdbcb! A B C D E Here s 4D nd 5D hypercube. Are you up to the chllenge?
9. The Complex Plne Abcb-Dbcb is written ll over the Mndelbrot set. The M-Set is fmous frctl with infinite vriety nd complexity, generted by repeting very simple rules. Mny free computer progrms re vilble to explore this fntstic structures. (These were mde with the freewre progrm Xos). Zooming in on the nose to the fr left revels one bcb pttern s shown here there re mny more you cn find.
10. Music Abcbdbcb sounds like it could represent series of notes. Indeed, if bcbdbcb is plyed on pino, the result is beutiful nd hunting. Downlod copy from bcbx.com. There is lso fully orchestrted version vilble for free downlod. 11. www.abcbx.com More Abcb vilble online for FREE Red Mggie nd the Abcb Genies Downlod Clssroom Activities, Worksheets, nd Blckline Msters Downlod Abcb Music Cheers! mike@bcbx.com Abcbx
Roly-Annie Roly-Annie is solitire crd gme nmed fter the crd queen of Mississippi riverbots, Roly Annie Keim. To ply, use the crds ce through seven. Mke the odd numbered crds blck nd the even numbered crds red, nd stck them in order so tht when the stck is fce up, the ce is on top. You re llowed to mke two dditionl stcks in this gme, right stck nd left stck. You re llowed to move the top crd of ny stck (on the tble or in your hnd) to the top of ny other stck (on the tble or in your hnd). However, you my move only one crd t time, nd you my never plce higher numbered crd on top of lower numbered crd; you my only plce lower number on higher number (the ce hs vlue of 1). You my ply ny crd on n empty stck. The object of the gme is get ll of the crds into either the left or right stck. The strt of gme is shown below. A A 2 A 2 A AA432 A AA43 A AA43 2 A 3 2 A??? 3 A AA4 A AA4 1. Wht is the minimum number of moves required to win this gme? 2. List t lest the first 20 moves. (You my wish to devise your own nottion.) 3. Describe ny ptterns you found while investigting this gme. Describe strtegy tht will llow you to win every time without mking mistke. 4. You nd friend decide to rce. You split deck of crds nd number the crds in ech hlf from 1 to 26. Plying by the sme rules s before, bout how long would it tke to decide winner?
POP-UP FRACTAL This eye-popping frctl is big hit with kids (nd grown-ups, too)! It s esy to crete something complicted nd beutiful using ides of frctls. The following pge contins the pttern for mking this work of rt. Supplies: Copies of the hndout on the next pge Scissors optionl: 9 x 12 construction pper nd glue sticks 1. Hve students fold the pge in hlf long the dotted line. Fold so tht the print is on the outside of the folded pge. 2. Cut long the two hevy line segments tht connect to the crese nd fold the flp upwrds. The corners of the flp will fit netly in the L brckets printed on the pge. 3. Open the pge, reverse the fold in the center of the flp nd close the pge so the flp is now completely (nd netly!) inside the folded pper. 4. Two move hevy lines re now seen to rech the creses in the center of the pper. Cut long these lso, then crese, using the L brckets to help. After folding the flps up, you cn flip the pper over nd fold the flps in the opposite direction. This will mke it esier to reverse the flps nd pop them inside. 5. When you ve popped the two smller flps inside, you should now be redy to mke two more cuts long the hevy lines tht now meet the four creses. Continue in this mnner. Point out tht the first cuts mde one flp, the second cut mde two flps, nd the third mde 4. How mny flps re mde by the finl cut? When it is done, you will hve beutiful 3-D pop-up form. This cn be glued to folded sheet of construction pper to mke lovely piece of rt or n eye-popping greeting crd. Questions for discussion: How is this frctl? How is this self-similr? How mny of ech size block re there? Wht other properties cn you investigte? Wht do you notice when you look t your model from different perspectives?