Definition:Kakeya's Constant
(Redirected from Definition:Bloom-Schoenberg Number)
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Definition
Kakeya's constant is defined as the area of the smallest simple convex domain in which one can put a line segment of length $1$ which will coincide with itself when rotated $180 \degrees$:
- $K = \dfrac {\paren {5 - 2 \sqrt 2} \pi} {24} \approx 0 \cdotp 28425 \, 82246 \ldots$
This sequence is A093823 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
This article is complete as far as it goes, but it could do with expansion. In particular: Needs considerable work done here by someone who understands exactly what is going on here. The case of the equilateral triangle is well known; so is the case of the Perron tree; I also remember a piece by Martin Gardner on the subject which demonstrates that a star-shaped area derived from the deltoid can be made arbitrarily small; and so on. Exactly what is meant here by simple convex domain needs rigorous clarification. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also known as
Kakeya's constant is also known as the Bloom-Schoenberg number, for Melvin Bloom and Isaac Jacob Schoenberg.
Also see
Source of Name
This entry was named for Soichi Kakeya.
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,28425 82246 \ldots$
- Wisewell, Laura and Weisstein, Eric W. "Kakeya Needle Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KakeyaNeedleProblem.html