Lohmann International Associates
ローマンインターナショナルアソシエイツ
176-0021東京都練馬区貫井
カナダの年金アクチュアリー、USエンロールドアクチュアリー、社会保障制度による各種の給付コンサルタント。
Tokyo, JAPAN
Leslie John Lohmann, FSA, FCIA, FNZSA
Canadian Pension Actuaries, United States Enrolled Actuaries and Employee Benefit Consultants Serving the International Community
Note: only the firm name and description are in Japanese.
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Last Updated January 25, 2006 (note that references are updated independently.)

Copyright © 2006 Lohmann International Associates All Rights Reserved
Pentagon inscribed in a circle of radius 'Disney' Reading Dan Brown's The DaVinci Code got me very interested in the irrational real number phi. Recalling that a pentagon could be inscribed within a circle using a compass and straight edge, I proceeded to recall how to do it.

Our forefathers did some pretty great stuff - including showing the world how to do this. Without showing the graphics (available elsewhere in cyberspace), suffice it to say that, beginning with a circle of radius 1, one draws a right triangle with one leg of length 1 (conveniently, the radius) and the other of length .5 (gosh - half the radius!). The hypotenuse has length sqrt (5)/2 (thanks Pythagoras). One then draws another right triangle with one leg of length 1 (the radius, again) and the other of length (sqrt (5)-1)/2. The length of the second leg you will recognize as the irrational real number often identified by the Greek letter "little phi." I personally prefer calling this number phi, instead of its inverse (often called Phi, using the capital Greek letter Phi and also equal to 1+phi. Capital Phi is the length of the line segment from the edge of the circle throught the origin to the mark used to draw the second triangle. Little phi, of course, is the length of the same line segment past the origin to the same mark.).

The second right triangle has a hypotenuse equal to the length of the sides of the inscribed pentagon sqrt(2-phi).

Back to Dan Brown.

In The DaVinci Code he suggests that Walt Disney used a lot of symbols in plain sight. He offered several examples from movies. Walt Disney Star inscribed in a circle

Without comments on beliefs or intentions, I offer here observations about the most important Disney symbol - the classic "Walt Disney" signature used as a logo. The one I used may be seen at disney.go.com/disneypictures/downloads. It is the one with "Walt Disney" in front of the Cinderella castle. Note that the logo does not have a space between "Walt" and "Disney;" WaltDisney.

1. The "dot" over the "i" in Disney is capital Phi as typically drawn by hand.
2. The terminating "y" in Disney is little phi as typically drawn by hand.
3. The capital Phi is "above the line" and equal to 1 divided by little phi - extending below the line.
4. Capital Phi minus "1" (the lower part of the "i" in Disney) equals little phi.
5. One might also see a capital Phi in the "D" of Disney.
6. Without much imagination, one could also suggest that the "W" in Walt is another little phi.
7. Same for the "t" in Walt.
The golden sections of 'WaltDisney' 8. If the length of the entire logo is capital Phi (1+little phi), the "Disney" portion (beginning precisely on the edge of the "bump" on the left of the D) is length 1 and the "Walt" portion (the rest) is length phi. In other words, "Disney" is phi of the horizontal length of the whole logo and "Walt" is phi squared of the whole logo; the Golden Section. 'Disney' is a Golden Section all by itself.  So is the 'D' all by itself. As you can see, the "Disney" portion split between the "D" and the "i" is also in Divine proportion. Finally (is it? 'Walt' is a Golden Section all by itself.  So is the 'W' all by itself. 'Walt' is a Golden Section all by itself.  So is the 'W' all by itself. So is the 'isney' all by itself. ), the right side of the downstroke of the "D" itself is the golden mean of the "D!"
How perfect is it?
The 'Walt Disney' logo - apparently perfect in every 'proportion.'
Decide for yourself!
9. If one created a right triangle with legs equal in length to "Walt" (to the edge of the "D") and "Disney," the This is the Waltograph TTF. We are in PHI (capital PHI)/phi (little phi) heaven! hypotenuse would be precisely equal to the length of the side of the inscribed pentagon in a circle of radius of length "Disney."

Incidentally, if you would like to see a lot of capital and little phi's, check out the "Walt Disney" font ("waltograph").

Copyright © 2006 Lohmann International Associates All Rights Reserved
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