Kakeya's 1917 problem: turn a unit needle continuously through a full rotation, staying inside a region — which region has the smallest area? Watch the needle turn in the classic solutions, from the disk down to the sliver-thin star, its swept positions painting the region. A companion to the Besicovitch-sets applet (the arbitrarily-small multiply-connected set). Constructions from Pál (1921), Besicovitch (1963), and Cunningham–Schoenberg (1965).
The needle (thick, one red end) turns while its swept positions (faint) fill the Kakeya set. Runs entirely in your browser; no network.
One historical construction is not yet included here, for want of the source material: H. J. van Alphen, “Uitbreiding van een stelling van Besicovitsch,” Mathematica: Tijdschrift voor studeerenden voor de acten wiskunde M.O. en voor studeerenden aan universiteiten, Afdeeling B, jaargang 10 (1941/42), pp. 144–157 (W. J. Thieme & Cie, Zutphen) — which showed a Kakeya set of arbitrarily small area can be kept inside a bounded circle. The article has proved very hard to find online; if you have a digitized copy, I would be glad to obtain it and add the construction to this app.