exploring and discovering mathematicsexploring and discovering mathematics references richard p....

Exploring and Discovering Mathematics


Joel Reyes [email protected]

http://joelnoche.multiply.com/

Department of MathematicsCollege of Arts and Sciences, Ateneo de Naga University

8:00 AM–10:00 AM, July 30, 2012WIRED talks, St. Joseph School, Naga City

Camarines Sur, Philippines

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Overview

Exploring mathematicsWho wrote The Royal Book of Oz?Order in Pollock’s chaos

Discovering mathematicsFolding paper in half repeatedlyTriangle inequalities and triangle centers

An open problemDo all triangles have periodic billiard paths?

The slides for Exploring mathematics are based on the talk Mathematics in the Artsand Sciences given on June 17, 2008 at the Ateneo de Naga University (AdNU), whichwas based on the talk The Role of Mathematics in the Total Development of theHuman Person given on February 9, 2008 at AdNU.

The slides for An open problem are from the talk Periodic Billiard Paths in Trianglesgiven on February 4, 2012 during the 2012 Bicol Mathematics Conference at AdNU.

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Exploring mathematics

Who wrote The Royal Book of Oz?

Who wrote The Royal Book of Oz?[Binongo, 2003]

L. Frank Baum wrote The WonderfulWizard of Oz in 1900. He died in 1919while writing the 14th Oz book Glindaof Oz. The 15th Oz book The RoyalBook of Oz, was supposedly written byBaum and ‘enlarged and edited’ by RuthThompson. By 1939, Thompson hadpublished the 33rd Oz book. In 2003,Binongo used multivariate analysis andstylometry to resolve the dispute of whoreally wrote The Royal Book of Oz.

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Preparing the data setFinding 50 variables to describe each text

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Choosing the words to study

Fifty stylometric variables (word, average rate of occurence in %)

the 6.7 with 0.7 up 0.3 into 0.2 just 0.2and 3.7 but 0.7 no 0.3 now 0.2 very 0.2to 2.6 for 0.7 out 0.3 down 0.2 where 0.2a/an 2.3 at 0.6 what 0.3 over 0.2 before 0.2of 2.1 this/these 0.5 then 0.3 back 0.2 upon 0.1in 1.3 so 0.5 if 0.3 or 0.2 about 0.1that/those 1.0 all 0.5 there 0.3 well 0.2 after 0.1it 1.0 on 0.5 by 0.3 which 0.2 more 0.1not 0.9 from 0.4 who 0.3 how 0.2 why 0.1as 0.7 one/ones 0.3 when 0.2 here 0.2 some 0.1

Function words (and not content words) were chosen.Auxiliary verbs and personal pronouns were not included.Misspellings were corrected and contractions were expanded.

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Reducing 50 dimensions into 2

Each book was partitioned into blocks of 5 000 words.

This yielded a matrix of 223 text blocks and 50 words.

To help us visualize this, the multivariate statistical technique ofprincipal component analysis was used.

The best two-dimensional approximation is found by rotating theoriginal 50 axes to new axes so that the latter represent directionsof decreasing variability.

The 50 principal component (PC) scores are approximated by thefirst two PC scores. (The 1st PC accounts for 20% of the totalvariation; the 2nd PC 7%.)

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Baum’s and Thompson’s canonical works (Oz-related)

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Baum’s and Thompson’s canonical works (Oz-related)Component loadings

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Baum’s non-canonical works (Oz-related)

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Baum’s non-canonical works (not Oz-related)

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Baum’s and Thompson’s short stories

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Glinda of Oz (1920), The Royal Book of Oz (1921)

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Visitors from Oz by Gardner (1998)

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Visitors from Oz by Gardner (1998) (continued)

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Order in Pollock’s chaos

No. 5, 1948 by Jackson Pollock[Vogel, 2006]

In 2006, this painting wasreportedly sold for about$140 million, the highestsum ever known at thattime to have been paid fora painting.

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Fractal analysis of Pollock’s drip paintings[Taylor, Micolich, & Jonas, 1999, Taylor, 2002]

Scientific objectivity proves to be an essentialtool for determining the fundamental contentof the abstract paintings produced by JacksonPollock in the late 1940s. Pollock dripped paintfrom a can onto vast canvases rolled out acrossthe floor of his barn. Although this unorthodoxtechnique has been recognized as a crucial ad-vancement in the evolution of modern art, theprecise quality and significance of the patternscreated are controversial. Here we describean analysis of Pollock’s patterns which shows,first, that they are fractal, reflecting the finger-print of nature, and, second, that the fractaldimensions increased during Pollock’s career.

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Fractals are self-similar[Taylor, 2002]

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The fractal character of Pollock’s paintings[Taylor, 2002]

The painting is scanned into acomputer. It is separated intoits different colored patterns,then covered with a computer-generated mesh of identicalsquares. The computer analyzeswhich squares are occupied andwhich are empty. This is donefor different mesh sizes. Thepatterns were found to be frac-tal over the entire size range.

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Studying the paintings chronologically showedthat the complexity of the fractal patterns, D,increased as Pollock refined his technique.

One D value is clearly an outlier—1.9 in 1950,a work that Pollock later destroyed. He mayhave thought this image was too dense or toocomplex and subsequently scaled back.

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Discovering mathematics

Folding paper in half repeatedly

How to fold paper in half twelve times[Historical Society of Pomona Valley, 2011, Weisstein, 2012]

The loss function for folding paper inhalf was derived by high school stu-dent Britney Gallivan in 2001. Brit-ney then set a new world record byfolding first gold foil and then paperin half 12 times in 2002, thus debunk-ing the assertions of Math@Home andPBS Kids that paper cannot be foldedin half more than eight times.

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Triangle inequalities and triangle centers

New triangular inequalities in the form of s – a, s – b, and s – c[Tsai, 2010]

In 2010, 17-year-old Kang Ying Liu ofHawai’i discovered nine new geometric for-mulas for describing triangle inequalities.For this, she took first place in the 53rdHawai’i State Science and Engineering Fair.

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Triangle inequalities and triangle centers

New triangle centers associated with a triad of simulated circumcircles[Bigold, 2011, Kimberling, 2012]

In 2011, Kang Ying Liu discovered threenew triangle centers. For this, she took firstplace in the 54th Hawai’i State Science andEngineering Fair.

Two of these triangle centers have beennamed after her.

(The Encyclopedia of Triangle Centers has called tri-angle centers X(3598) and X(3599) the 1st and the2nd Liu points, respectively.)

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An open problem

Do all triangles have periodic billiard paths?

The slides can be found athttp://www.adnu.edu.ph/bmc2012/Noche.pdf

The paper can be found athttp://www.adnu.edu.ph/bmc2012/bmc2012.pdf

A one-page summary can be found at http://www.scribd.com/doc/36872171/Periodic-Billiard-Paths-in-Triangles

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http://www.adnu.edu.ph/bmc2012/Noche.pdf

http://www.adnu.edu.ph/bmc2012/bmc2012.pdf

http://www.scribd.com/doc/36872171/Periodic-Billiard-Paths-in-Triangles

http://www.scribd.com/doc/36872171/Periodic-Billiard-Paths-in-Triangles


It’s your turn

It’s your turn

Look for articles and books by mathematics popularizers such asMartin Gardner and Ian Stewart. Also look at Danica McKellar’sbooks and her website at

http://www.danicamckellar.com/

More examples of high-school students who have made originalcontributions in mathematics can be found at

http://math.stackexchange.com/questions/174009/

Other easily-understood open problems in mathematics can befound at

http://mathoverflow.net/questions/100265/

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http://www.danicamckellar.com/

http://math.stackexchange.com/questions/174009/

http://mathoverflow.net/questions/100265/


References

Pat Bigold.

St. Andrew’s Priory girls dominated state science/engineering fair.HawaiiNewsNow, April 4, 2011. Retrieved July 29, 2012 from http://metrooahu.hawaiinewsnow.com/

news/schools/st-andrews-priory-girls-dominated-state-scienceengineering-fair/50306

Jose Nilo G. Binongo.

Who wrote the 15th book of Oz? An application of multivariate analysis to authorship attribution.Chance, 16(2):9–17, 2003.

Historical Society of Pomona Valley.

Folding paper in half twelve times.December 7, 2011. Retrieved July 29, 2012 from http://pomonahistorical.org/12times.htm

Clark Kimberling.

Encyclopedia of Triangle Centers.April 17, 2012. Retrieved July 29, 2012 fromhttp://faculty.evansville.edu/ck6/encyclopedia/ETCPart2.html

Richard P. Taylor.

Order in Pollock’s chaos.Scientific American, 287(6):116–121, 2002.

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http://metrooahu.hawaiinewsnow.com/news/schools/st-andrews-priory-girls-dominated-state-scienceengineering-fair/50306

http://metrooahu.hawaiinewsnow.com/news/schools/st-andrews-priory-girls-dominated-state-scienceengineering-fair/50306

http://pomonahistorical.org/12times.htm

http://faculty.evansville.edu/ck6/encyclopedia/ETCPart2.html


References

Richard P. Taylor, Adam P. Micolich, and David Jonas.

Fractal analysis of Pollock’s drip paintings.Nature, 399:422, 1999.

Michael Tsai.

Hawaii teen whiz creates new math formulas.The Honolulu Advertiser, April 21, 2010. Retrieved July 29, 2012 fromhttp://the.honoluluadvertiser.com/article/2010/Apr/21/ln/hawaii4210335.html

Carol Vogel.

A Pollock is sold, possibly for a record price.The New York Times, November 2, 2006. Retrieved July 29, 2012 fromhttp://www.nytimes.com/2006/11/02/arts/design/02drip.html?_r=1&adxnnl=1&ref=arts&adxnnlx=

1163031599-revbGMuaIhdTP4qLonq8BA&oref=slogin

Eric W. Weisstein.

Folding.From MathWorld–A Wolfram Web Resource. Retrieved July 29, 2012 fromhttp://mathworld.wolfram.com/Folding.html

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http://the.honoluluadvertiser.com/article/2010/Apr/21/ln/hawaii4210335.html

http://www.nytimes.com/2006/11/02/arts/design/02drip.html?_r=1&adxnnl=1&ref=arts&adxnnlx=1163031599-revbGMuaIhdTP4qLonq8BA&oref=slogin

http://www.nytimes.com/2006/11/02/arts/design/02drip.html?_r=1&adxnnl=1&ref=arts&adxnnlx=1163031599-revbGMuaIhdTP4qLonq8BA&oref=slogin

http://mathworld.wolfram.com/Folding.html

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