Say it with Numbers, but NOT Prime Numbers

Just read an article summary from the Journal of Marketing Research on a twitter tip by Alex Bellos about the preference for composite numbers in product names... Her is most of the good stuff. 

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A series of experiments documents the influence of numbers on the liking of brands. For example, an imaginary brand name for anti-dandruff shampoo (Zinc) is more liked when it includes a common product number (e.g., Zinc 24) than when with includes a prime number (e.g., Zinc 31). The research also shows that the presence of the operands responsible for the sum or product further enhance the liking of a brand name. For example, not only is a Volvo S12 more liked than a Volvo S29, but liking is further enhanced when an advertisement for a Volvo S12 includes a license plate with the numbers 2 and 6. The operands 2 and 6 make 12 more familiar because they encourage the subconscious generation of the number 12.
The influence of available operands on liking for the number brand extends to advertising claims. The authors conducted a study in which consumers were asked to make a choice between V8 and Campbell’s tomato juice. Some consumers saw a V8 advertisement that stated, “Get a full day’s supply of 4 essential vitamins and 2 minerals with a bottle of V8” whereas other saw an advertisement that stated “Get a full day’s supply of essential vitamins and minerals with a bottle of V8.” More consumers chose the bottle of V8 when the number 4 and 2 were explicitly mentioned in the claim. Creating similar advertisements for Campbell’s tomato juice did not influence preferences for Campbell’s.

More on Pi and the 47 Ronin

A few years ago I wrote this post... Since then I have learned enough to add an addendum at the bottom...
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If you take the Asakusa Line from Shinagawa, just one stop away you will come to one of the most famous shrines in all of Tokyo, the Sengakuji Temple. It isn't the biggest, prettiest, or most ornate, but it is rich with the kind of history the Japanese love. This is the resting place of the 47 Ronin, one of Japan's most popular samurai stories.

"The story has all the elements for a Hollywood production: a good, noble guy who dies unfairly; a corrupt court official and cunning villain who is disliked by everyone but seems to be always ahead of the game; the good guy’s loyal subordinates who are totally determined to avenge their master’s death at whatever price, even with their own lives... In the end, the story has sparked the imagination and inspired the utmost respect from an entire nation for over 300 years."[Luis Estrada's Travel Blog]

I came across a mathematical reference to the story in a March 1908 article in the American Mathematical Monthly I received recently from Dave Renfro.

"In Tokyo, at the Buddhist temple of Sengakuji lie buried the forty-seven Ronin, the national heroes of feudal Japan. Just within the gate, in a two-storied building, swords, armor and other relics of these heroes are shown on payment of a fee. By the side of the path leading to the tombs is a well with the inscription, 'Here they washed it.' No one in Japan needs to be told that it was the bloody head they were bringing to the grave of their lord, that dead master for whom they considered it the highest privilege thus to forfeit all their lives. The popular reverence for these heroes is still attested not only by the incense perpetually kept burning before their tombs but in stranger fashion by the fresh visiting cards constantly left upon their graves. [To someone who is still there, do the Japanese still leave these visiting cards?]"

"All the world knows their exploit, but who knows that one of them, Shigekiyo Matsumura, was the greatest Asiatic mathematician of his age, who in his work Sanso, published in 1663, calculated the length of one side of a regular inscribed polygon of 32768 or 2¹⁵ sides, obtaining 0.000095873798655313483 and thence for the value of pi 3.141592648, which is accurate to seven places of decimals, to eight significant figures..."

I would be thrilled if any of the folks who still read this in the Tokyo area would send a digital picture of the tomb of Matsumura so that I can add it to this note.
ADDENDUM -----------------------------------
The very kind Arjen Dijksman connected me with Japanese Physicist Tasuo Tabata who gave me some detail.  It seems that  Matsumura might be more appropriately called Shigekiyo Muramatsu.  He did write the sanso, and all the math things described seem to be a modest description of his contributions.  However, Professor Tabata tells me that he was NOT one of the 47 Ronin.  He is, however, connected to the story. The professor says, "Shigekiyo had only a daughter. Her husband Hidenao and their son Takanao joined the 47 ronin. --"... So now, I guess I'm down to wanting a picture of the 47 Ronin from Sengakuji Temple. And if anyone knows where the grave of Shigekiyo Muramatsu is located, and/or has a picture I would love to have one.
Professor Tabata has written the details he found in slightly more detail here.

Thanks again to professor Tabata and Arjen for their help.

Gauss' Missing Blackboard

I just came across a document on the web that I can't find the source of but it has a really nice story about Gauss and the constructable 17-gon story. It seems to be related to someone named Alex Anderson...(if you know that person, he has a picture I want a copy of very much)..
"Gauss was visiting Braunschweig (Brunswick) and still lying in bed on 29 Mar 1796 when he realised the connection of the cyclotomic polynomial to the construction of the n-gon and how to construct the 17 gon. Ahrens [p. 11] says Gauss sent the slate on which he had done the calculation to Wolfgang Bolyai, who preserved it." Does anyone know where that slate is today?

Tangent Sequences of a Cubic

Old mathematicians never die;
they just go off on a tangent.


Recently re-introduced to a pretty property of the tangents to cubics from Ross Honsberger's book, More Mathematical Morsels (Dolciani Mathematical Expositions).  I will illustrate a simple part of the property, then provide a source for a really nice extension that seems to be pretty new.

As the magician says, take a cubic, take any cubic... OK, so graph any y=f(x) where f(x) is a third degree polynomial. (you really don't have to graph it, but it might help you see a second property).  

1)  Pick any point other than the inflection point and record the x-value as A

2) Write the equation of the tangent line at that point
3)  Find where the tangent line intersects f(x) , call this B
4)  Repeat the process starting at Point B to get a point C,
5)  Do it again starting at C to get point D...
Look at the sequence A, B, C, D... What do you observe?


Ok, so let's practice a little integration. Find the area between the tangents and the curve.  Now think of the ratio of the two areas big/little...
Focus deeply, I'm reading your mind...
AHA, the ratio was 16.

"How does he do it?"

There is an additional nice extension about this idea explained by Alexander Bogomolny at his Cut-the-Knot web site.  Enjoy.