Algorithms and Approximations
Recall is a technical term in search. It refers to the (possibly
very many) terms returned. A counterpart is relevance. That term
points to the practical reality: many recalled items are not useful.
Abstraction and description are similarly paired terms.
The first can be thought of by referring to the following three line
drawings. Each presents some detail and in so doing conveys the notions
of a beach house, the Snake River, and Upper New York Bay.
Words - e.g., the names of distinct mathematical curves - present
another way to abstract some details and in so doing convey the essence
of a situation.
We could argue that visual and literal abstraction are just
ways to deal with some aspects of reality, and that quantity is another. These
things come together in some cases involving irrational numbers (and some
geometric issues about them).
Here is a brief statement of something that
could be viewed as an interesting open problem in mathematics and
(We) would like to see ... an algorithm that creates such good
approximations as either of these:
2^[(5/2)^(2/5)] = 2.718290995
||878/323 = 2.718266253869969
However the idea of a regular procedure leading to an approximation is
somewhat akin to having a blank page and moving from it to some drawing
like the detailed ones shown here.
The following visuals illustrate the pragmatic virtues of words and
mathematical concepts as ways to deal with physical properties. (Click
on any of the three images to see a larger version.)
Some Interesting Irrationals - and Rationals that Approximate Them
*Small print in drawing reads: "Upper New York Harbor from World Trade
Center Observatory, 6/23/87".
**Brewster, G.W. The Mathematical Gazette, 25-263, 49,
cited in Gaither, C. C. and Cavazos-Gaither, A. E.,
Speaking, A Dictionary of Quotations,
; the exponent for 2 is 5/2 raised to the 2/5.
***Ramanujan; cited by Gardner, M.
The golden ratio φ = (1+ √5)/2 = 1.61803...
The golden ratio conjugate Φ = 1/φ ≈ 0.6180339887
Φ/2 = (√5 - 1)/4 = sin 18 ° = sin π/10 ≈
(2sin π/10)(1+ √5)/2 ≈ 1
For further Exploration
consider adapting exposition to computer
capabilities. Or look at some Puzzles,
or the twelve elements in Pentaminoes.
[See Golomb, S., Polyominoes [with more than 190 diagrams by
Lushbaugh, NY: Charles Scribner's Sons, 1965]; Second Edition, Princeton
Princeton Univ. Press, 1994.]
©2013 Allen Klinger