Three Figures from a Vintage Treatise on Early Attempts to Get the Area of a Simply Connected ᐜ Kakeya Needle Set Below ⅛π
A ▘Kakeya needle set▝ is a delineated region of the Euclidean plane in which a unit line segment can be rotated continuously through a half-circle.
There are fabulously elaborate constructions for Kakeya needle sets of arbitrarily small area that ᐞare notᐞ simply connected ᐜ (See this post
https://www.reddit.com/r/mathpics/s/JKcsXGLZ5o
for somewhat about it) ... but for quite a long time the smallest known simply connected one was the thrain-becuspen hypocycloid – ie the shape generated by a point on a circle of radius of ¼ rolling on the inside of a circle of radius ¾ (or ᐞin-generalᐞ those radii scaled-up by aught @all ... but for the purpose of just permitting rotation of a unit line-segment inside it ᐞspecifically exactlyᐞ those radii) the area of which is ⅛π . But then, these three – & indeed successful – attempts to get it down a bit further came-about: the first one down to
2(π-1)/(π+8) ≈ 0.38443205028
, the second down to
(¹¹/₁₂-2log³/₂)π + ε ≈ 0.33218085595 + ε
, & the thriddie one down to
¹/₂₄(5-2√2)π + ε ≈ 0.28425822465 + ε
(cf. ⅛π ≈ 0.3926990817)
... with the second & third having that "+ ε" appent because the value it's appent to is what the area tends to as the number of asperities of the figure it pertains to tends to ∞ .
But in 1971 the goodly Dr Cunningham ᐞjust totally slewᐞ the problem with a mind-bogglingly complex construction that gets the area arbitrarily small, ᐞandᐞ that fits in a unit disc ... ᐞandᐞ - adding a very generous helping of double-cream & maple syrup on-top - ᐞis simply connectedᐞ ᐜ ! And yet: these early probings into the matter, & their associated figures, remain of great mathematical-historical interest. And they also have the charm about them of being instances of squeezing the very-last drop of juice out of relatively ordinary geometrical reasonings, before the juggernaut of fabulous excursionry into totteringly-lofted recursiferous edifices comes a-crashing into the scene.
ᐜ A 'simply connected' region of the Euclidean plane is one in which any closed path in it can be continuously contracted to a point: basically, it has no holes or handles, or any of that sort of thingle-dingle-dongle.
⚫
From
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ON THE KAKEYA CONSTANT
by
F CUNNINGHAM JR & IJ SCHOENBERG
¡¡ may download without prompting – PDF document – ¾㎆ !!
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It has long been known that K < ⅛π , this being the area of a three-cusped hypocycloid inscribed in a circle of radius ¾ . In 1952 R. J. Walker (7) determined by measurement the area of a certain set with a result that suggested that K < ⅛π . In spite of its heuristic value Walker's note has not become well known. Independently of it, but using the same general idea, A. A. Blank (3) exhibited recently certain star-shaped polygons with the Kakeya property, having areas approaching ⅛π , but not smaller than this value. Blank's examples suggested to each of us the possibility of finding Kakeya sets actually having areas smaller than ⅛π, each such set giving an upper estimate for K. In the present note three different kinds of such sets are described. The first two (Part I) are due to Cunningham, the third (Part II) to Schoenberg. Each of these examples is self-contained and may be read independently. They also furnish progressively better estimates, the third example showing that
(1)
K < ¹/₂₄(5-2√2)π = (0.09048 . . .)π .
After completing this paper we were informed that Melvin Bloom has also found the estimate (1) by exactly the same construction as described in Part II. Since he obtained (1) several months earlier than Schoenberg, the priority belongs to Professor Bloom.
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