The Lebesgue universal covering problem. Philip Gibbs code (The original code live here) to compute the area for many choices of rotation angle.
In 1914 Henri Lebesgue defined a 'universal covering' to be a convex subset of the plane that contains an isometric copy of any subset of diameter 1. His challenge of finding a universal covering with the least possible area has been addressed by various mathematicians: Pál, Sprague and H.C Hansen have each created a smaller universal covering by removing regions from those known before. However, Hansen's last reduction was microsopic: he claimed to remove an area of 6⋅10−18, but we show that he actually removed an area of just 8⋅10−21. In the following, with the help of Greg Egan, we find a new, smaller universal covering with area less than 0.8441153. This reduces the area of the previous best universal covering by 2.2⋅10−5.
The program is not very long. Please study it, add some test cases to this program or write your own, in your own favorite language!
The output of the program is here:
------------------ ----------------- ------------------
| Area | | Angle 1 | | Angle 2 |
------------------ ----------------- ------------------
0 0.8441281473635023 88.60789390240575 124.70567493165655
0.1 0.8441249398973186 88.71714220899656 124.56695879567842
0.2 0.8441221028463506 88.8260964414572 124.4285772267079
0.3 0.8441196363224983 88.93475925216121 124.29052749390299
0.4 0.8441175404406899 89.04313328141143 124.15280687881696
0.5 0.8441158153188825 89.15122115749935 124.0154126753377
0.6 0.8441144610780628 89.25902549676582 123.87834218962644
0.7 0.8441134778422481 89.36654890366236 123.74159274005537
0.8 0.8441128657384852 89.47379397081467 123.60516165714331
0.9 0.844112624896854 89.58076327908488 123.4690462834925
1 0.8441127554504655 89.68745939763818 123.33324397372115
1.1 0.8441132575354651 89.79388488400514 123.19775209440034
1.2 0.8441141312910305 89.90004228415069 123.062568023984
1.3 0.8441153768593765 90.00593413254083 122.9276891527414
1.4 0.8441169943857526 90.11156295220835 122.79311288269056
1.5 0.8441189840184462 90.21693125482504 122.6588366275253
1.6 0.8441213459087822 90.32204154076746 122.52485781254855
1.7 0.8441240802111265 90.42689629918976 122.3911738745986
1.8 0.8441271870828845 90.53149800809305 122.2577822619791
1.9 0.8441306666845043 90.63584913439738 122.12468043438602
2 0.8441345191794781 90.73995213401359 121.99186586283525
--------------------------------------------------------------------------
| Table of H.C Hansen values |
--------------------------------------------------------------------------
0 0.1339745962155614 4.952913815765206E-4
1 0.024131160666459442 2.418850424554875E-6
2 6.080990483914671E-4 3.750723412842599E-11
3 3.701744790810228E-7 8.454119457933276E-21
4 1.3702923282072605E-13 4.288332272808866E-40
5 1.87770106474412E-26 1.103387620921236E-78
6 3.5257612885412026E-52 7.304785452013354E-156
- Lebesgue’s Universal Covering Problem (Part 1 - 2013)
- Lebesgue’s Universal Covering Problem (Part 2 - 2015)
- Lebesgue’s Universal Covering Problem (Part 3 - 2018)
- Lebesgue's universal covering problem - Wikipedia
- Lebesgue's universal covering problem (ResearchGate)
- The Lebesgue universal covering problem (John C. Baez, Karine Bagdasaryan, Philip Gibbs - 2015)
- A lower bound for Lebesgue's universal cover problem (Peter Brass and Mehrnod Sharifi - 2005)
- An upper bound for Lebesgue’s universal covering problem (Philip Gibbs - 2018)
- Amateur Mathematician Finds Smallest Universal Cover (Popular article on this topic)
- Amateur Mathematician Finds Smallest Universal Cover (Hacker News)
- References related to Lebesgue's Universal Covering Problem