ALICE

    I have recently been reading a lot about the “Golden Ratio,” 1.61803 …, known by the Greek letter ϕ (pronounced “fee,” not “fye”), a mathematical ratio or proportion frequently found in nature by which many plants and animals grow and/or expand.  It is also manifest in certain architecture, art, music and science, and it has even been implicated in the behavior of the stock market!  The first known enunciation of it was by Euclid, the ancient Greek mathematician, who observed that any straight line cut at a certain point would manifest a ratio of the entire line to the longer segment as the longer segment is to the shorter segment.  It is known in German as the “Goldene Schnitt” (Golden Cut).

    The Ratio also represents the relationship of the numerical sequences in the “Fibonacci Series,” originally stated by Leonardo Fibonacci, a 16th Century Italian mathematician, who posited the Series about the reproduction of a pair of rabbits in captivity.  The Series is defined as each sequential number therein being the sum of the two preceding numbers in the Series, e.g., 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.  The Golden Ratio, an irrational number that does not repeat nor is evenly divisible by any integer, is also expressed in the design known as the “Golden Rectangle,” which is considered a rectangular proportion that is inherently pleasing to the human eye.  The ratio of the shorter side to the longer side calculates out to the Golden Ratio, and it is allegedly manifest in playing cards, index cards, the Parthenon façade, and many other examples.

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    Dr. Mario Livio published a book about The Golden Ratio.  Dr. Livio is an astrophysicist who, until 2015, was the director of the Space Telescope Science Institute, associated with the Hubble Telescope. Livio is discussing the nature of various historical number systems and how they used “bases” different than our own. Some cultures have used “base-5” (as with the abacus in China) and some even have used “base- 60” which, though awkward, probably accounts for the manner in which time and circles are divided into degrees, minutes and seconds. (Sixty just happens to be the lowest number evenly divisible by 1, 2, 3, 4, 5 and 6, as well as 10, and 12, 15 and 20.)

    Most Western cultures have used “base-10,” and the perfect example is a vehicle odometer.  Recalling my own elementary-school arithmetic, we learned to put numerals in columns, starting from right to left, as ones, tens, hundreds, thousands, etc. An odometer has a series of little “wheels,” with black numerals on a white ground, except that the rightmost “wheel” (showing tenths of miles) has white numerals on a black ground. The odometer, moving from right to left, thus shows on each “wheel” tenths of miles, miles, tens of miles, hundreds of miles, thousands of miles, etc.

   Livio discusses the nature of various historical number systems and how they used “bases” different than our own (which is “base-10”). Some cultures have used “base-5” (as with the abacus in China) and some even have used “base- 60” in the past which, though awkward, probably accounts for the manner in which time and circles are divided into degrees, minutes and seconds. (Sixty just happens to be the lowest number evenly divisible by 1, 2, 3, 4, 5 and 6, as well as 10 and 12.)

    Livio then cites a passage from Alice’s Adventures In Wonderland by “Lewis Carroll,” the pen name of Charles Dodgson, who lectured on mathematics at Oxford!  In an observation totally unrelated to the “Golden Section” Alice, agonizing over the strange things she has encountered, frets:

“I’ll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is 13, and four times seven is—oh, dear! I shall never get to twenty at that rate!”

    Crediting famous mathematician Martin Gardner’s The Annotated Alice, Livio points out that Alice’s bizarre math (4 x 5 = “12”) “works” IF “base-18” is used and would, therefore, be equivalent to our “20”: 1 (x 18) plus 2 left over! “1-2”! And 4 x 6 = “13” IF “base-21” is used, since our “24” may be thus expressed as 1 x 21 + 3 left over!

    So, I decided to play around with Alice’s (Dodgson’s) “system” and, recalling my elementary-school “times table” for “4,” I noted that every equation implied therein might be stated thus: 4 x 7 = “14”; 4 x 8 = “15”; 4 x 9 = “16”; 4 x 10 = “17”; 4 x 11 = “18”; 4 x 12 = “19”; 4 x 13 = 20; 4 x 14 = 21; 4 x 15 = 22; etc. In our base-10 system, those equations respectively “equal” 28, 32, 36, 40, 44, 48, 52, 56 and 60, but if the number “base” that Alice implicitly referenced is incremented by “3” for each succeeding equation (starting with base-21, then base- 24, base-27, base-30, base-33, base-36, base-39, base-42, base-45, etc.) then Dodgson’s equations can make sense! Consider: “4 x 7 = 14” works if “base-24” is used: “1-4” yields 1 x 24 + 4 “ones” left over = 28! “4 x 8 = 15” works if “base-27” is used: “1-5” yields 1 x 27 + 5 “ones” left over = 32! “4 x 9 = 16” works if “base-30” is used: “1-6” yields 1 x 30 + 6 “ones” left over = 36! “4 x 10 = 17” works if “base-33” is used: “1-7” yields 1 x 33 + 7 “ones” left over = 40! “4 x 11 = 18” works if “base-36” is used: “1-8” yields 1 x 36 + 8 “ones” left over = 44! “4 x 12 = 19” works if base-39” is used: “1-9” yields 1 x 39 + 9 “ones” left over = 48! “4 x 13 = 20” works if “base-42” is used: “2-0” (“1-0” + 10 “ones”) yields 1 x 42 + the 10 “ones” left over = 52! “4 x 14 = 21” works if “base 45” is used: “2-1” (“1-0” + 11 “ones”) yields 1 x 45 + 11 left over = 56! Whew! Mind-bending!

    That hidden math progression is rather intriguing, and though I have read the original Alice, I did not pick up on any of that until I had read Livio’s exposition. . It makes me wonder if there might be other math riddles in Alice In Wonderland.

    I think that the ORIGINAL version of Alice is also written on two levels: one for children and the other for adults. I think the “adult” version is a vicious satire on the dominant British culture of the day! I was quite amused while reading it. I also think that the Uncle Remus Tales by Joel Chandler Harris satirizes Southern US culture.

Most of us started with the Walt Disney versions of both in our childhoods.  However, I think Walt missed those subtle viewpoints. 

(8/16/17--updated 12/11/22; 4/29/24)