Buddha Loves Geometry

>> --GRADUATE STUDENT AT GRAND VALLEY-- >> YES. >> AND... WAYNE, WE'RE LOOKING FORWARD TO THIS. THANK YOU. >> WHOO-HOO! >> IT'S ALL YOURS. >> LET ME GIVE MYSELF A HAND TO MYSELF FIRST. (clapping) I'LL PUMP MYSELF UP. (audience laughing) ALL RIGHT-- ALL RIGHT, SO, YEAH. SO, I KNOW MOST OF THE INSTRUCTORS HERE I'VE EITHER HAD AS TEACHERS BEFORE OR I'VE KNOWN YOU GUYS SINCE I WAS HERE. I WAS A STUDENT HERE FROM '96 TILL, UM, WINTER OF '99. SO, IT'S BEEN A WHILE SINCE I'VE BEEN HERE, BUT I'VE BEEN WORKING HERE SINCE MARCH OF 2000. AGAIN, THAT'S WHEN I GOT LAID OFF. BUT I LIKE THIS-- THIS IS WHAT I DO. I LOVE TEACHING MATH AND TUTORING MATH, AND SO... ANYWAYS, OKAY, AND-- SO, DO ANY OF YOU GUYS KNOW ME A WHILE-- KNOW ME, UH, FOR A WHILE THERE. I LIKE TO CRACK JOKES, SO... AND THIS IS ABOUT THE BEST BUDDHA PICTURE I CAN FIND ON INTERNET, SO I FIGURE-- (audience laughing) THIS-- THAT'S IT. ALL RIGHT! SO "BUDDHA LOVES GEOMETRY... AT LEAST ACCORDING TO JAPANESE," NOW... WELL... JAPAN'S NOT THE ONLY COUNTRY THAT, YOU KNOW, BELIEVES IN BUDDHISM, BUT SOMEHOW, THIS IS WHAT I FOUND. ALL WE'RE GONNA TALK ABOUT IS WE'RE GONNA TALK ABOUT AN OBJECT-- IT'S CALLED A "SANGAKU." AND THAT'S THE JAPANESE CHAR-- UH, KANJI FOR "SAN" AND "GAKU." LITERALLY, IT MEANS, UH, "CALCU"-- UH, THIS IS "CALCULATION" AND THIS IS FOR-- I'M NOT QUITE SURE WHAT THAT IS, AND I COULDN'T FIND IT, BUT IN GENERAL, THAT MEANS "YOU'RE GONNA CALCULATE." SO... AND SANGAKU, NOW-- WHAT THEY ARE IS THEY ARE PAINTED ON WOOD PIECES. SO-- AND SO, AND THOSE WOOD PIECES, NOW, THEY'RE SAYING, "WELL, YOU KNOW WHAT? "WE GOT THOSE MATH PROBLEMS WE'RE GOING TO PAINT ON WOOD," AND THEY BROUGHT IT TO TEMPLES TO HANG INSIDE THE TEMPLES. SO-- AND LITERALLY WHAT THEY DO IS THEY ACTUALLY OFFER THOSE TO THE BUDDHIST GODS. IN BUDDHISM, THERE ARE TONS OF THEM-- THERE ARE MANY, MANY GODS. I MEAN, THERE'S GODS FOR, LIKE, FOR YOUR STOVE. THERE'S GODS FOR YOUR HOUSE. THERE'S GOD FOR THE LAND. THERE'S GODS FOR LOTS OF THINGS. SO, THEY TAKE THOSE AS OFFERINGS. AND ALSO, THEY USE THOSE AS KIND OF THE CHALLENGE FOR THE PEOPLE THAT GO TO THE TEMPLE. SO, AND THERE ARE MOST-- AND MOST OF THEM WERE FOUND IN THE EDO PERIOD, BETWEEN THOSE TIMES, AND AFTER-- NOW, IN 1968, WHEN THE EDO PERIOD ENDS, THAT'S WHEN THEY STARTED INDUSTRIALIZATION, BECAUSE, UM-- (mic cuts out) --ARE DESTROYED AS WELL. SO FAR, THERE ARE ABOUT 900 KNOWN THAT'S LEFT. AND BEFORE, THERE WERE LOTS AND LOTS OF 'EM. SO, IT'S JUST A BRIEF HISTORY, AND THAT'S WHAT THE SANGAKUS LOOK LIKE. SO, YOU GOT YOUR TEMPLE, AND THERE'S SOME OF THE PIECES IN THERE, AND I TRIED TO GET, LIKE, A PIECE OR TWO OF THESE TOGETHER, BUT HERE ARE SOME OF THE QUESTIONS, IF YOU CAN SEE UP ON THE PICTURES. AND THESE ARE THE QUESTIONS THAT THEY HAVE FOR EACH OF THE PICTURES. AND I'M SURPRISED THAT THEY ACTUALLY LASTED THIS LONG BECAUSE IT'S PAINT, YOU KNOW? ESPECIALLY ON WOOD. I FIGURED IT'D GET WEATHERED. AND HERE'S MORE-- AND NOTICE, UH, HERE'S THE CHARACTERS FOR THE SANGAKU, RIGHT HERE, AND YOU GOT YOUR DIFFERENT SHAPES-- CIRCLES, LITTLE FANS, PART OF THE CIRCLE, AND THERE INVOLVES A TRIANGLES ON THIS ONE... AND YOUR DIFFERENT SQUARES. AND THIS, ACTUALLY, RIGHT HERE, THIS PERSON ACTUALLY WROTE DOWN THE DATE OF THIS, IS WHEN THIS WAS PAINTED AND THIS PERSON'S NAME... BUT I CAN'T REALLY READ ANY OF THAT STUFF THAT'S UNDERNEATH THERE. IT'S JUST-- I JUST CAN'T READ IT. SO, OKAY. SO, HERE'S ANOTHER BETTER-LOOKING PIECE. SO, THERE ARE MORE, DIFFERENT TYPES OF PROBLEMS. OKAY, SO THIS IS THE ONE THAT WE'RE GOING TO TACKLE TODAY. SO, WHAT YOU SEE HERE IS YOU HAVE A RIGHT TRIANGLE-- I WASN'T ABLE TO GET THAT RIGHT ANGLE SIGN IN THERE. BUT YOU HAVE A RIGHT TRIANGLE, AND YOU HAVE, UH... THREE CIRCLES THERE. THE BIG CIRCLE TANGENT TO ALL THREE SIDES. THE MIDDLE CIRCLE TANGENT TO THE BASE AND THE HYPOTENUSE, AND THE BIG CIRCLE. AND THE SMALL CIRCLE IS TANGENT TO BOTH SIDES AND THE MIDDLE CIRCLE. SO, THAT'S WHAT WE'RE GIVEN. SO, NOW, ADDITIONAL CONDITIONS-- SO, THE BASE IS TWICE AS LONG AS ITS HEIGHT. SO, I'M GOING TO ACTUALLY DRAW A BETTER PICTURE OVER HERE. WELL, MAYBE NOT BETTER. SO, 2X-- HERE'S X. AND I ADD MY CIRCLES. SORRY. OKAY. AND THE THREE CIRCLES ARE TANGENT. AND SO, ALL THE TANGENTS THAT YOU TALKED ABOUT. AND THE GOAL IS THAT WE NEED TO FIND THE RADII OF ALL THREE CIRCLES. AND ADDITIONAL TOPIC AFTER THAT IS THAT, SUPPOSE WE KEEP THE PATTERN GOING, "WHAT'S THE SUM OF THE AREA OF ALL THE CIRCLES?" SO, IMAGINE WE HAVE MORE CIRCLES, KEEP GOING. AND SO ON. OKAY. NOW, BEFORE WE CAN ACTUALLY-- ACTUALLY, WE TALK ABOUT FINDING THE RADIUS. I WAS STUMPED-- BECAUSE I'M LIKE, "OKAY, THIS DOESN'T REALLY TELL ME A WHOLE LOT." I HAVE THREE CIRCLES WITH TWO MEASUREMENTS OF 2X AND X. THERE'S NO NUMBERS WHATSOEVER. SO, I WAS LIKE, "WHAT DO I DO?" I MEAN, THE MOST I CAN THINK IS, "OKAY, WELL, I HAVE A RIGHT TRIANGLE," SO A RIGHT TRIANGLE-- WHAT DO YOU GUYS THINK, FIRST THING? IF YOU HAVE A RIGHT TRIANGLE? >> (indistinct speaking). >> AND ALSO, IF YOU HAVE A DISTANCE OF TWO SIDES AND YOU HAVE A RIGHT TRIANGLE, THE FIRST THING YOU THINK OF IS WHAT? >> THE HYPOTENUSE. >> HYPOTENUSE. SO, PYTHAGOREAN THEOREM, RIGHT? OKAY, SO I THINK ABOUT PYTHAGOREAN THEOREM AND I SAY, "ALL RIGHT, SO A^2 + B^2 = C^2," SO I HAVE THIS. OKAY, THIS IS MY HYPOTENUSE. ALL RIGHT, NOW I LOOK AT IT. WELL, IT STILL DOESN'T REALLY TELL ME A WHOLE LOT, BECAUSE IF I WANT TO FIND THE RADIUS OF ALL THESE THINGS... BUT NOW, REMEMBER, IN GEOMETRY CLASS, WE ACTUALLY TALK ABOUT SOMETHING CALLED THE "INCENTER." THE INCENTER OF A TRIANGLE IS THAT... IF YOU HAVE A CIRCLE THAT IS TANGENT TO ALL THREE SIDES-- WHICH IS WHAT WE HAVE-- THAT PARTICULAR CENTER IS CALLED AN "INCENTER." SO, BASICALLY, MY GIVEN CONDITION THAT A CIRCLE IS TANGENT TO ALL THREE SIDES-- SO, THIS RIGHT HERE IS MY INCENTER... AND INCENTER, NOW-- IF YOU HAVE AN INCENTER, THAT MEANS THAT THE RADII FROM THE INCENTER IS ALSO PERPENDICULAR TO... THE SIDES OF THE TRIANGLE. LET ME GET A DIFFERENT COLOR. OKAY. SO, THIS IS WHAT I HAVE. SO, I WAS THINKING ON THE LINE OF THIS. OKAY. THIS TOOK ME THREE HOURS TO FIGURE IT OUT. (laughing) I WAS STARING AT IT FOR A WHILE. AND THEN, I REALIZED, "OH! "OKAY, SO..." NOW, IF I WANT TO-- NOW, ONCE I HAVE ALL THE SIDES AND I KNOW THIS, WELL, AT LEAST I CAN FIND THE AREA OF THE TRIANGLE, RIGHT? BASED ON ITS HEIGHT-- TIMES IT-- DIVIDED BY 2? OKAY, BUT ANOTHER THING IS NOW I REALIZE, "WELL, IF I HAVE... "THE RADIUS COMING UP FROM THE INCENTER, I COULD DO THIS." AND THIS IS WHAT HAPPENS OVER HERE. I HAVE JUST SPLIT-- THIS IS A RIGHT TRIANGLE INTO THREE DIFFERENT TRIANGLES. A-B-O, B-O-C, AND C-O-A. THOSE ARE MY THREE TRIANGLES. AND SINCE THE RADII-- THEY'RE ALL PERPENDICULAR TO THE BASE, TO EACH SIDE-- SO, WHAT I CAN DO IS I CAN FIND THE AREA OF A-B-O PLUS THE AREA OF B-O-C PLUS THE AREA OF A-O-C, AND THAT SHOULD BE THE SAME AS ONE-HALF TIMES X TIMES 2X. OOPS, DROPPED. AND THIS IS EXACTLY WHAT I DID. SO, HERE IS SOME OF MY DESCRIPTIONS. OKAY, SO... AND I WANT TO APOLOGIZE AHEAD OF TIME. I DID THIS IN OFFICE 2010, AND WHEN I REDID IT AGAIN ON 2007, SOME OF THE FUNCTIONS GOT LOST. SO, THE-- AND I COULDN'T FIX IT-- I CAN'T FIX THE SLIDE FOR THAT REASON. SO, THIS IS BASICALLY WHAT I DID. I FIRST FOUND THE AREA OF EACH... SO, I HAVE... 1/2 X TIMES R1 + 1/2 (2X) TIMES R1 AND + 1/2 THE HYPOTENUSE-- THE BASE-- TIMES ANOTHER R1. SO, R1 X TIMES SQUARE ROOT OF 5. AND THAT'S GONNA EQUAL TO MY AREA OF 1/2 TIMES THE BASE TIMES HEIGHT. 2X^2. OKAY, AND FROM HERE, WE KNOW THIS-- WE GOT A BUNCH OF HALVES FOR EACH TERM AND THE Xs IN EACH TERM, SO WE CAN BASICALLY JUST CROSS ALL THOSE OUT. AND THE RESULT IS THAT WE HAVE... R1 TIMES (1 + 2 + SQUARE ROOT OF 5) = 2X. SO, IN THIS CASE, DIVIDING 3 + SQUARE ROOT OF 5, ON BOTH SIDES, WE HAVE R1 IS 2X OVER 3 + SQUARE ROOT OF 5. OKAY. SO, ANY QUESTIONS SO FAR? SO, PRETTY STRAIGHTFORWARD, RIGHT? >> YEAH. >> OKAY, SO NOW, THIS IS WHERE THE FUN BEGINS. OH, YEAH, SO... OKAY. SO, NOW, THE PROBLEM IS-- WELL, OKAY, SO I KNOW THIS RIGHT HERE-- NOW, WE HAVE ALL THE VALUES FOR THE R1s. NOW, HOW AM I GONNA FIGURE OUT R2? WELL, WHAT WE CAN DO HERE, IS I NOTICE-- WELL, GUESS WHAT? THE-- FROM THE INCENTER OF A (indistinct)-- TO DROP THIS STRAIGHT DOWN, TO R1, TO THIS POINT, IF I GO ALL THE WAY ACROSS, WHAT HAPPENED HERE IS YOU NOW HAVE A RIGHT TRIANGLE... THAT CONTAINS R2, AND EVENTUALLY, R3. SO, NOW, LEARNING-- REMEMBER FROM TRIG... WE CAN USE SIMILAR TRIANGLES. SO, BEFORE I FIND R2, ANOTHER PROBLEM IS WE REALLY DON'T KNOW THAT ALL THE CENTERS LAY ON THE SAME LINE. YOU JUST DON'T KNOW-- YOU CAN'T ASSUME. BUT... BUT WHAT WE CAN DO HERE IS, WELL, IF YOU THINK WITH ME, RIGHT HERE... IF I DROP A VERTICAL LINE RIGHT HERE WHERE THE R2-- ON THE EDGE OF R2... AND SAME THING FOR R3... WHAT HAPPENED HERE IS I HAVE SIMILAR TRIANGLES, BECAUSE YOU COULD DROP A VERTICAL LINE, THESE TWO LINES ARE PARALLEL, SO THEREFORE, THEY SHARE THE TWO SIDES OF THE SAME ANGLE. SO, THEY'RE SIMILAR. SO, THEREFORE-- WITH MORE WORK, OBVIOUSLY-- BY THAT, WE CAN SAY THAT THE CENTERS WILL BE LAYING ON THE SAME LINE, AS WELL. SO, THEREFORE, WE CAN USE-- SO, THE SIMILAR TRIANGLE ARGUMENT WILL BE VALID. OKAY? SO... SO, THAT'S BASICALLY WHAT WE'RE LOOKING AT RIGHT NOW. OKAY, SO... SO, THIS IS WHAT WE HAVE RIGHT HERE. THIS IS R2. R1, WE SAID IS... 2X OVER 3 + SQUARE ROOT OF 5. ALL RIGHT, SO, NOW, THE PROBLEM IS, WELL, I NEED TO FIGURE OUT WHAT THIS SIDE IS AND WHAT-- OOPSY, I'M SORRY. I NEED TO FIGURE OUT WHAT THIS SIDE IS AND WHAT THIS SIDE IS. NOT ENOUGH INFORMATION, RIGHT? LOOKS LIKE IT, BUT WE'LL MAKE IT WORK. SO... OKAY. YEAH, SEE, THIS IS WHAT HAPPENS. NOW, THE THING IS-- OKAY, ORIGINALLY, NOW REMEMBER, WE SAID THE BASE OF THE BIG TRIANGLE IS 2X, CORRECT? SO, SINCE THIS IS R1 RIGHT HERE-- SO, THAT MEANS THE DISTANCE FROM THIS POINT TO THIS POINT, GOES ARE R1. SO, I COULD FIND FROM THIS POINT TO THIS POINT BY SUBTRACTING IT FROM 2X. AND THAT'S EXACTLY WHAT I WILL DO, AS SOON AS I GET THIS TO WORK. THERE YOU GO. SO, WE TAKE 2X - (2X OVER 3 + SQUARE ROOT OF FIVE) AND WE HAVE THIS DISTANCE RIGHT HERE. SO, HERE'S THAT. NOW, OKAY. AND SO, FROM HERE, AGAIN, WE HAVE A RIGHT TRIANGLE. SO, PYTHAGOREAN THEOREM ONCE MORE, AND WE CAN FIND OUT THE HYPOTENUSE OF THIS LENGTH RIGHT THERE. THIS... AND SO, WE JUMP THROUGH ALL THE HOOPS, DOING THIS. THERE YOU GO. SO, A^2 + B^2 = C^2, AND DID ALL THE OTHER WORK. AND SO, WE ARRIVED AT THE HYPOTENUSE IS NOW 2X (SQUARE ROOT OF 10 + 4 (SQUARE ROOT OF 5)) DIVIDED BY 3 + SQUARE ROOT OF 5. I KINDA LIKE THIS. YOU KNOW THIS-- YOU HAVE THE 3+ SQUARE ROOT OF 5 ALL OVER THE PLACE, WHICH IS REALLY COOL. OKAY, SO... NOW, THE BIGGEST PROBLEM IS, WELL, WHAT'S GONNA HAPPEN HERE? IF YOU REMEMBER FROM THIS PICTURE, NOTICE... FROM THE FIRST CENTER AND TO THE SECOND CENTER, YOU HAVE TWO-- IT'S BASICALLY ONE LINE SEGMENT MADE OUT WITH WHAT? R1 AND R2, RIGHT? SO, WHAT WE'RE GONNA HAVE TO DO HERE, IS WE'RE GONNA HAVE TO SUBTRACT THAT OUT. SO, THAT MEANS... SO, THAT MEANS I NEED TO TAKE-- TO FIGURE OUT THIS DISTANCE, I HAVE TO TAKE MY NEWLY FOUND HYPOTENUSE MINUS R1 AND MINUS R2. SO, THIS IS WHERE IT GETS REALLY MESSY. BAM. YES. WHAT I DID HERE WAS-- WELL, I ACTUALLY-- SO, RIGHT HERE, ON THE LEFT-HAND SIDE, LOOK INSIDE ON THE WAY TOP, THAT'S WHERE YOU HAVE THE HYPOTENUSE MINUS R1 AND MINUS R2, AND... SO, THAT'S THIS SEGMENT RIGHT HERE. SO, BASICALLY, WHAT I HAVE IS TWO TRIANGLES, AGAIN, USING SIMILAR TRIANGLES. OKAY, SO, BASICALLY, I USED SIMILAR TRIANGLES AND USE MY PROPORTION, SO HERE'S MY ORIGINAL HYPOTENUSE DIVIDED BY THE R1 EQUAL TO THE SHORTER HYPOTENUSE DIVIDED BY R2. AND ONE OF THE FIRST THING HERE IS THAT I ACTUALLY TOOK CARE OF THE RIGHT-HAND SIDE FIRST, SINCE THEY HAVE THE SAME DENOMINATOR. AND I SIMPLIFY IT DOWN TO THIS. AND FROM HERE, I CROSS-MULTIPLY. SO... AND ONCE WE CROSS-MULTIPLY, WHAT WE CAN DO HERE IS THAT-- WELL, IF YOU REMEMBER SOME OF THE-- FOR SOME OF THE ALGEBRA STUDENTS, RIGHT? YOU HAVE ALL THE-- ESPECIALLY IN TRIG, YOU HAVE Xs ON ONE SIDE, WITH THE REALLY UGLY COEFFICIENT. SO, WE BRING ALL THE VARIABLES ON ONE SIDE, WE FACTOR IT OUT. SO, I MOVE THIS R2 TO OVER HERE, FACTOR OUT THE R2, AND DIVIDE IT BY-- DIVIDE IT BY THE REST. SO, NOW WE HAVE THIS (indistinct). IT'S-- THIS TOOK ME A WHILE BECAUSE I WAS-- I'M DECENT AT ALGEBRA, BUT I'M STILL NOT PERFECT. NOBODY CAN BE PERFECT, SO I MADE A FEW MISTAKES IN MIDDLE, WHILE I'M MULTIPLYING. AND ONE MISTAKE COST ME, BECAUSE IT'S-- I THINK I MADE A MISTAKE SOMEWHERE AROUND-- I THINK AROUND HERE, WHEN I WAS MULTIPLYING STUFF. AND IT WAS BAD. IT TURNED OUT-- I THINK I ENDED UP GETTING A NEGATIVE NUMBER FOR RADIUS. (audience laughing) AND THAT, OBVIOUSLY, DOESN'T WORK. SO, OKAY! SO, ANOTHER THING IS-- ALL RIGHT, SO-- AND FROM HERE, I WAS LIKE, "OKAY, WELL, IF YOU LOOK AT THIS"-- DO YOU SEE ANYTHING GOOD IN R2? SOMETHING THAT LOOKS AT LEAST FAMILIAR? >> (indistinct speaking). >> THAT'S YOUR R1, ISN'T IT? OKAY, SO THAT MEANS, BASICALLY, WE'RE SAYING HERE-- R2 IS... R1 TIMES WHATEVER THAT IS. SO, WE HAVE THE... THAT WAS A MINUS 1 AND THIS IS THE SAME THING. PLUS ONE. SO... ALL RIGHT, SO I FOUND-- AND I WAS LIKE, "ALL RIGHT, MY IMAGINATION'S KINDA OKAY, YOU KNOW?" SINCE THESE FOLLOW SOME KIND OF PATTERN, SO HOPEFULLY, WE HAVE US A PATTERN OF SOME SORT, RIGHT? SO, I WAS THINKING, "WELL, IF THIS R1, "AND R2 IS BASICALLY R1 TIMES THIS," SO (indistinct) WHAT? R2 WILL BE-- R3 WILL BE WHAT? YOU MULTIPLY BY IT AGAIN, RIGHT? >> (indistinct speaking). >> RIGHT-- WELL, OF COURSE. WE CAN'T JUST ASSUME, SINCE WE'RE MATHEMATICIANS. SO, WHAT WE GONNA DO HERE IS WE'RE GONNA DO THIS. WE'LL FIND THE R3, AND... ACTUALLY, IT'S GONNA BE SIMILAR TECHNIQUES, AS WELL. BECAUSE WHAT HAPPENED HERE IS... SO, NOW, IF I'M LOOKING AT MY R2 AND R3-- SO, AGAIN... IS R2 AND R3. SO, WE'RE BASICALLY GONNA APPLY THE SAME TECHNIQUE HOW I DID OVER HERE. AND THAT IS GOING TO BE VERY LONG. (chuckling) YOU GOT-- IT'S JUST BECAUSE THE NUMBER IS SO UGLY AND IT GOT HUGE, REALLY, REALLY QUICK. AND SO, SAME IDEA, RIGHT? I TAKE MY RESULT... AND MINUS R2 AND R3-- DIVIDED BY R3, SO THERE IS MY SMALLER TRIANGLE RIGHT HERE. AND ON THE LEFT, THAT'S ACTUALLY THE-- USING THE BIGGER TRIANGLE FROM HERE TO HERE, SO... OKAY. SO, RIGHT HERE, AGAIN, WE'RE GONNA-- WE'RE GONNA CROSS-MULTIPLY. (laughing) ALL RIGHT, AND ACTUALLY, I BELIEVE I SKIPPED A STEP, BECAUSE WHAT HAPPENED HERE? YOU NOTICE ON THE LEFT. AGAIN, THEY'RE THE SAME DENOMINATOR, CORRECT? SO, WHAT WE COULD DO IS WE COULD TAKE CARE OF THAT RIGHT THERE. SO, IT'S LESS WORK. SO, YOU CROSS-MULTIPLY, THEN WE HAVE THIS BIG... GLOB OF NUMBERS AND EXPRESSIONS... AND SO, WE TOOK CARE OF THAT. AND SIMPLIFY MORE, WE HAVE THIS. AND... OKAY, SO... SO, WE MULTIPLY, AND THEN, WE AGAIN-- WE, UM-- SO, WHAT HAPPENED HERE IS THAT DIVIDE THIS-- THIS PARENTHES OF A FRACTION ON BOTH SIDES. AND LUCKILY, SOME OF 'EM DO CROSS OUT. BUT IT ACTUALLY TOOK ME FOR A WHILE, BECAUSE I'M DECENT AT FACTORING BUT NOT WITH ANYTHING THAT INVOLVES RADICALS. ESPECIALLY WITH RADICALS, NOW, BECAUSE YOU'RE GONNA END UP HAVING-- SAY, IF YOU MULTIPLY THIS, RIGHT? YOU HAVE 10 + 16 TIMES 5, AND IT'S NOT LIKE YOUR NORMAL X AND Ys. IT'S JUST HARDER TO DO. AND IT ACTUALLY TOOK ME A WHILE TO REALIZE, "OH, BY THE WAY, "WHEN I SIMPLIFY ALL OF THIS, I ACTUALLY GET THIS BACK." AND GUESS WHAT? PATTERN FOLLOWS. SO... AND FROM HERE, I ACTUALLY DID THE MORE WORK TO CALCULATE WHAT R4 IS, BUT R4 IS-- IT GETS WORSE. IT GETS WORSE AND WORSE EVERY TIME YOU DO IT. OKAY. SO, BASICALLY, SO FAR, WHAT WE FOUND IS THAT, WELL, WE KNOW R1 IS... 2X OVER (3 + SQUARE ROOT OF 5). AND WE SAID R2 IS... BASICALLY, 2X OVER (3 + SQUARE ROOT OF 5), TIMES... (SQUARE ROOT OF 10 MINUS (4 SQUARE ROOT OF 5,)) MINUS 1, OVER (SQUARE ROOT OF 10 MINUS (4 TIMES SQUARE ROOT OF 5)) + 1. >> YOU MEAN PLUS 4? >> OH, I'M SORRY. IT'S PLUS-- THANK YOU. COULDN'T SEE IT-- THANK YOU VERY MUCH. SO-- AND THAT JUST MEANS THAT R3, THEN, WE FOUND OUT THAT WHAT THAT BASICALLY IS-- WE JUST KEEP GOING. OKAY. ALL RIGHT, SO-- SO, THAT MEANS IF YOU DO R3, WE'RE BASICALLY GONNA-- WE'RE GONNA KEEP THE SAME 2X + 3 + SQUARE ROOT OF 5... AND NEXT ONE IS 1, 2-- THEN, WE'RE GONNA DO THE SAME THING FOR-- SO, R4'S JUST ONE LESS. SO, WE'RE GONNA KEEP GOING ON OUR PATTERN. OKAY, SO-- ANY-- ANY QUESTIONS... ABOUT HOW WE COME UP WITH THE R1, R2, AND R3? AND IF IT DOESN'T MAKE SENSE, PLEASE LET ME KNOW AND I'LL DO MY BEST TO EXPLAIN IT FOR YOU. OKAY. ALL RIGHT, SO, NOW, THE THING IS-- WELL, IF YOU LOOK LIKE-- IF YOU LOOK AT THIS, NOW, IT SEEMS LIKE, BASICALLY... WE'RE MULTIPLYING THE RADIUS BY THE SAME THING OVER AND OVER AGAIN, RIGHT? SO, IF YOU'RE IN CALC-- IF YOU'RE IN, YOU KNOW, CALC II OR IN 110-- YOU GUYS REMEMBER TALKING ABOUT SERIES? RIGHT, INFINITE SERIES? SO, IF YOU'RE MULTIPLYING THE SAME NUMBER OVER AND OVER AGAIN, IT SOUNDS LIKE A GEOMETRIC SERIES TO ME, RIGHT? SO, THIS IS WHAT WE'RE GONNA DO THEN. SO, WE'RE GONNA THINK OF THIS AS AN INFINITE SERIES... TO WHERE WE'RE LOOKING FOR THE AREA OF ALL THE CIRCLES. AND SO, THIS IS WHAT WE HAVE. SO, BASICALLY, MY FIRST CIRCLE IS GONNA BE PI TIMES R1 SQUARED, PI TIMES R2 SQUARED-- WE'RE GONNA JUST KEEP THINKING THAT WAY. THEN-- NOW, THE THING THAT WE REALIZE THAT WE MADE THE CONNECTION SAYING NOW, "WELL, R2, R3, AND R4-- THEY'RE BASICALLY JUST R1 MULTIPLIED BY SOME NUMBER." AND SO, THIS IS WHAT WE'RE GONNA DO. FIRST THINGS FIRST. AND WE'LL FACTOR THAT PI OUT... 'CAUSE THERE'S A PI THERE. NOW, WHAT WE'RE GONNA DO HERE IS-- WELL, WE'RE GONNA REWRITE THE R2 AND R3 AND R4 IN TERMS OF R1. BECAUSE WE KNOW THIS, THEN... WE HAVE ALL THAT FUN STUFF. SO, THAT MEANS R2-- I RE-WROTE THAT IN TERMS OF R1, JUST LIKE WHAT WE HAD OVER HERE. AND SAME THING FOR R3, AS WELL, AND SO ON. NOW, SINCE WE HAVE EVERYTHING IN TERMS OF R1, ESPECIALLY R1 SQUARED, WHAT I CAN DO IS I CAN FACTOR THAT OUT FROM EVERY SINGLE TERM. RIGHT, SO FACTOR EVERYTHING OUT, SO NOW, I HAVE 1 + THIS SQUARE ROOT PLUS-- NOW, THIS IS RAISED TO A FOURTH POWER. OKAY, SO... AND NOW, THE THING IS, IF YOU LOOK AT IT NOW, WHAT HAPPENED HERE IS, WELL, YOU HAVE-- NO, YOU HAVE THIS PARENTHESES, YOU HAVE 1 + THIS QUANTITY SQUARED PLUS THIS QUANTITY TO THE FOURTH, DOESN'T REALLY LOOK LIKE A GEOMETRIC SERIES, BUT HOWEVER, IF YOU ACTUALLY MULTIPLY EVERYTHING OUT, WHAT YOU HAVE HERE IS THAT... OR YOU COULD THINK OF IT THIS WAY, AS YOU MULTIPLYING BY WHAT YOUR "R" IS... JUST-- NOW, TO REFRESH EVERYBODY'S MEMORY ON THE GEOMETRIC SERIES... BEFORE I DO THAT. IT'LL BE... SO, THIS IS ALL GEOMETRIC SERIES. AND SO, WHAT WE COULD DO HERE IS THAT, IF-- WE CAN TREAT THIS AS-- WE CAN TREAT OUR "R" AS-- IN THIS CASE, "R" AS THE SQUARE OF-- SO, IT-- THIS IS GOING TO BE OUR "R" IN THIS CASE. IF YOU KNOW THIS, WE'RE MULTIPLYING BY THIS AT EVERY SINGLE TIME. AND NOW-- THE ONLY PROBLEM IS NOW, IF I WANNA FIND THE SUM OF THIS, THE "R" HAS TO BE LESS THAN 1, CORRECT? SO, THEN, IF YOU CAL-- IF I ACTUALLY CALCULATE THIS AND-- THIS IS INDEED LESS THAN ONE... BECAUSE IF YOU LOOK AT IT, YOU HAVE THE SAME VALUE, BUT ONE'S MINUS ONE AND ONE'S BIGGER THAN ONE. SO, THEREFORE, THE NUMERATOR HAS TO BE SMALLER THAN THE DENOMINATOR. THEREFORE, THE WHOLE FRACTION'S LESS THAN ONE. SO, WHEN YOU SQUARE, "R" HAS TO BE LESS THAN 1. YOU DON'T HAVE A CHOICE. SO-- SO, THEN, WHAT WE CAN DO IS, THERE IS A FORMULA FOR FINDING THE SUM OF A GEOMETRIC SERIES... ESPECIALLY THE INFINITE ONES. LET'S CALL... "A" OVER (1 - R), WHERE "A" IS YOUR FIRST TERM. SO, THAT MEANS, IN OUR CASE, WHAT WE COULD DO IS, OUR FIRST TERM IS GONNA BE 1, AND WE'RE GONNA DO 1 MINUS 1-- 1 OVER (1 - R), WHICH, IN THIS CASE, WE HAVE THIS BIG FRACTION RIGHT HERE. AS I'VE SHOWN-- HERE IT IS. OKAY. SO, FINALLY... WE HAVE THAT. (laughing) UM, SO... SO, THIS BEAUTIFUL FRACTION OF 11 + (4 SQUARE ROOT OF 5) + 2 TIMES 10 + (4 SQUARE ROOT OF 5)-- (sighing) DIVIDED BY THE (indistinct), 4 TIMES SQUARE ROOT OF 10 + (4 SQUARE ROOT OF FIVE) IS THE AREA-- IS A SUM OF THE AREA OF ALL THE CIRCLES IN THIS PARTICULAR TRIANGLE. BUT, OF COURSE, THEN, AT LEAST, THAT'S IT FOR THE RADIUS. AND ONCE YOU CALCULATE OUT THE PI TIMES R1 SQUARES, IT DEPENDS ON WHAT THE X VALUE IS. AND THEN, THIS WILL GIVE YOU THAT-- SORRY, THE ACTUAL SUM OF THE ENTIRE TRIAN-- ENTIRE CIRCLE IN THIS PARTICULAR TRIANGLE. SO... IT'S-- IT REALLY TOOK ME A WHILE AND I STUMBLED ON THIS MANY, MANY, MANY TIMES, JUST BECAUSE THE ALGEBRA WAS JUST-- IT WAS JUST A LOT OF ALGEBRA, AND I CAN'T-- I JUST COULDN'T-- I COULDN'T DO IT IN THE FIRST TIME. IT TOOK ME PROBABLY ABOUT-- OH, COLLECTIVELY-- MAYBE BETWEEN TEN AND FIFTEEN HOURS. AND DO JUST A LITTLE BIT AT A TIME, BECAUSE I CAN'T FOCUS. I JUST CAN'T FOCUS AT ALL. SO... SO, ANY QUESTIONS? >> WAS THIS ONE OF THE ONES THAT WAS ACTUALLY FOUND AS A PAINTING IN ONE OF THE TEMPLES? >> ACTUALLY, I HAVE NO WAY OF VERIFYING IT. SO, IT DON'T REALLY HAVE, LIKE, A DATABASE OF WHAT'S FOUND AND WHAT'S NOT. >> AND JUST BY THE LOOKS OF THE THINGS THAT YOU WERE SHOWING ME, IT LOOKS LIKE THEY HAD A LOT OF PROBLEMS THAT INVOLVED CIRCLES? >> YES, A LOT OF CIRCLES, A LOT OF TRIANGLES. JUST DIFFERENT SHAPES. THEY HAVE JUST-- EVEN SOME INVOLVE SQUARES OR RECTANGLES. THEY JUST THROWIN' 'EM ON. THERE WERE A LOT OF 'EM. SO... YES, ANYTHING ELSE? YES, MR.HIRSCH? >> SO, THESE DATE BACK TO... 1600s? >> YES, 1600s AND ON, YES. >> DO YOU HAVE ANY IDEA HOW THE JAPANESE AT THAT TIME WOULD HAVE SOLVED THESE? I MEAN, ARE THEY USING-- WHAT YOU ALL ARE SEEING HERE IS MODERN ALGEBRAIC EQUATIONS. >> RIGHT. >> NOW, AT THE END OF THAT PERIOD THAT EXISTED IN EUROPE. I DON'T KNOW WHAT THE JAPANESE HAD. AT THE BEGINNING OF THAT PERIOD, THE MODERN ALGEBRAIC EQUATION WOULDN'T HAVE EXISTED ANYWHERE. SO, DO YOU HAVE ANY IDEA WHAT THEY ACTUALLY DID TO SOLVE THESE PROBLEMS? >> MMM, I ACTUALLY TRIED TO LOOK INTO THIS AND SEE IF THERE WAS SOME KIND OF, LIKE, A SOLUTION TO IT. LIKE, HOW THEY WOULD PRESENT A SOLUTION, BUT I WAS NOT ABLE TO FIND ONE. SO-- BUT I THINK-- BUT-- I CAN READ SOME OF THE JAPANESE CHARACTERS, AT LEAST ACCORDING TO SOME OF THOSE TABLETS. AND SO, I ASSUME THEY WOULD PROBABLY HAVE EVERYTHING WRITTEN. MAYBE NOT NECESSARILY IN TERMS OF X OR Y, BUT MAYBE THEY'RE JUST SAYING, "THIS IS SOMETHING," IN JAPANESE. SO, THIS-- AND SO, JUST USE THAT INSTEAD OF THE X AND Y NOTATION-- OR "R" AND "A"-- THAT WE USE. SO... AND THIS IS ACTUALLY REALLY INTERESTING BECAUSE, YOU KNOW-- I MEAN, JAPAN IS-- YOU KNOW, IT'S-- THEY'RE A COUNTRY ON JUST-- THEY'RE SURROUNDED BY THE SEA, SO I DON'T THINK THEY'RE GOING TO HAVE A WHOLE LOT OF COMMUNICATION WITH THE GUYS UP OVER IN EUROPE. (audience chuckling) SO, STUFF THAT-- I MEAN, A LOT OF TIMES, THERE'S NO-- ALL-- I MEAN, NOT EVEN JUST IN ***-- ASIA AND EUROPE, BUT EVEN JUST WITHIN THE COUNTRIES, A LOT OF TIMES YOU HAVE MATHEMATICIANS. THEY'RE DOING THE SAME THING, BUT INDEPENDENTLY. LIKE NEWTON AND... (indistinct speaking). CALCULUS. YES? WHO--SORRY. SAY YOURS. >> DID YOU HAVE THE... WORD FOR THAT CONSTANT, ANYWHERE ON THE INTERNET, SEE IF THAT CONSTANT SHOWS UP ANY PLACE ELSE? >> (sighing) UM, I DID NOT. I ACTUALLY DID NOT. BUT YEAH, THE THING IS, ALL OF A SUDDEN, WE HAVE THIS. IT JUST KIND OF SHOWS UP, ALL OF A SUDDEN. SO... YES, SIR? >> WOULD YOU SAY THAT THESE QUESTIONS WERE COMPOSED TO OUR EQUIVALENT OF GRADUATE LEVEL OR HIGH SCHOOL LEVEL OR-- >> THIS ONE? UM... WELL, THE THING IS, I DON'T THINK-- UH, THE IDEA IS-- THE IDEA OF THE GEOMETRY NOW-- I THINK THAT'S A HIGH SCHOOL LEVEL GEOMETRY, BECAUSE I DON'T THINK INCENTER IS NOT-- THAT'S NOT SOMETHING THAT'S REQUIRED GRADUATE LEVEL. I MEAN, I CAN-- I CAN-- I'M PRETTY SURE I'VE HEARD THAT BEFORE, IN THE GEOMETRY CLASS. BUT IT'S-- I THINK IT'S THE-- AND ALGEBRA IS NOT NECESSARILY HARD, IT'S JUST LONG AND HAS A LOT OF TERMS. IT JUST GETS MESSY. SO, I THINK HIGH SCHOOLERS CAN DO IT. BUT MAYBE, YOU JUST NEED TO BUCKLE DOWN AND JUST LOOK AT IT SEVERAL TIMES, INSTEAD OF JUST TRYING TO DO EVERYTHING IN ONE SHOT. 'CAUSE I CERTAINLY COULDN'T DO IT. SO-- >> BUT, NOW, THESE THINGS WERE IN TEMPLES, THOUGH, RIGHT? >> YES. >> SO, IT WAS MORE OF A MEDITATIVE KIND OF THING? I MEAN, IT WAS NOT POSED TO STUDENTS. IT WAS-- >> RIGHT. THAT'S WHY THEY'RE SAYING-- THEY WERE OFFERING-- THEY WERE PUTTING THESE IN THE TEMPLE AS OFFERING TO GODS. SO, IT'S LIKE-- I DID NOT KNOW BUDDHA LIKES GEOMETRY. I REALLY DIDN'T KNOW. (audience laughing) SO-- YES, MR. WARRINGTON? >> COULD YOU GIVE US SOME OTHER EXAMPLES OF WHAT WAS ON THE WALL, OTHER THAN THIS ONE? >> UM... YEAH, THEY HAD SOME, LIKE-- EVEN SOME EXAMPLES. I WONDER IF I CAN FIND IT. NOW, I WASN'T ABLE TO FIND-- THERE WERE A FEW OTHERS, BUT I DIDN'T LOOK IT UP. BUT I BELIEVE THERE WERE, IF I GO BACK TO THIS SLIDE RIGHT HERE, I THINK THIS WOULD PROBABLY BE A GOOD ONE TO LOOK AT. YEAH, SO THERE YOU HAVE A FAN SHAPE WITH A SIMILAR CIRCLES, AND... YEAH, AND SOME OF THESE WRITINGS-- THIS IS THEIR LANGUAGE. I MEAN, I CAN READ THE CHARACTERS, BUT THE LANGUAGE IS A LITTLE BIT OLD. SO, I'M NOT EXACTLY UNDERSTAND WHAT THEY'RE TALKING ABOUT. SO... YEAH, IT MENTIONED ABOUT THE DIFFERENT CIRCLES THAT YOU HAVE AND THE FAN SHAPE AND THE-- BUT I JUST COULDN'T MAKE OUT WHAT THEY'RE SAYING. SO, IF I CAN FIND, MAYBE, LIKE, A TRANSLATION, WE CAN PROBABLY DO IT. ANYTHING ELSE? YES? >> YOU SAID THAT THESE WERE FOUND INSIDE OF TEMPLES. WERE THEY KIND OF, YOU KNOW-- IF YOU COME TO THE TEMPLE, YOU COME AND DO THESE FOR FUN? OR WAS THIS SOMETHING THAT, YOU KNOW, IS REQUIRED OF YOU BEFORE YOU ENTER OR EXIT THE TEMPLE? IS IT A LESSON? >> UM... (laughing) (audience laughing) >> IF YOU WERE A BUDDHIST, THEY'D BE ABLE TO SOLVE THIS. (general chatter) YOU'D SIT THERE FOR, YOU KNOW, 45 DAYS. (audience laughing) >> IF YOU CANNOT DO THESE MATH PROBLEMS, YOU'RE EXCOMMUNICATED FROM THE BUDDHA TEMPLE. (audience laughing) NO, NO, NO. UM, NO-- ACTUALLY, THEY SAID THAT THIS IS ACTUALLY A PART FOR FUN, AS WELL. SO... >> IT'S COOL FUN. (audience chuckling) >> WELL, YOU KNOW, HEY, WHATEVER FLOATS YOUR BOAT. (audience chuckling) OKAY, ANYTHING ELSE? YES-- YES, SIR? >> SO, WAYNE, YOUR CURIOSITY AS A MATHEMATICIAN-- WHAT HAPPENS IF YOU CHANGE THE 2X TO MX? >> 2X TO MX? UM... SO, THAT MEANS I GET TO-- >> I JUST WONDERED IF YOU EXPLORED THAT-- >> NO, I HAVE NOT, ACTUALLY. >> OKAY. >> WELL, I COULD PROBABLY DO IT. >> THIS NEXT WEEK. (audience laughing) >> YEAH-- NO, I'M SURE-- I'M SURE IT'S DOABLE. I MEAN, IT'S JUST THAT-- THERE'S SOMETHING-- THE CALCULATIONS ARE GONNA HAVE TO BE ADJUSTED WHEN THE 2 BECOMES AN "M," AND WE'RE JUST GOING TO HAVE A LOT MORE TERMS. >> YOU WONDER WHAT THE COMMON RATIO'S GONNA BE AND HOW IT RELATES TO ONE AS A SOLID, AND IT'S AN INTERESTING PROBLEMS HERE. >> YEAH. YEAH, IT JUST IS REAL-- THIS THING REALLY TOOK ME FOR A WHILE, BECAUSE WHEN I FIRST-- WHEN THE DR. NOVOTNY AT GRAND VALLEY FIRST HANDED OUT THIS PROBLEM, I WAS LIKE, "THERE IS NO WAY-- HOW DO YOU EVEN DO THIS? "ARE YOU KIDDING ME?" BUT YEAH-- BUT AFTER DOING A LOT OF-- A LITTLE BIT OF DIGGING-- ACTUALLY, A LOT OF DIGGING, AND THROUGH OLD INFORMATION I FOUND EARLIER WEEKS IN THE CLASS, AND WE'RE LIKE, "OH, YEAH." EVERYTHING JUST KIND OF LEAPS FROM ONE TO ANOTHER. SO, I MEAN-- YEAH, THIS THING'S PRETTY MUCH-- NO, I MEAN, THIS THING GOES FOR WITH OTHER MATH CLASSES, AS WELL, I THINK. I'M TAKING "NUMBER THEORIES" RIGHT NOW, AND A LOT OF STUFF WE'RE DOING RIGHT NOW IS ALL DEFINITION, DEFINITION-- PRIOR DEFINITIONS. AND YEAH. OKAY, ANY OTHER QUESTIONS? ALL RIGHT, THEN, THANK YOU VERY MUCH FOR COMING. (applause) >> THANKS, WAYNE.