Saturday, April 21, 2018

The High Steppe

Drugs are not new to the human experience.
[T]he Yamnaya people, who swept out of Central Asia about 5000 years ago and left their genes in most living Europeans and South Asians, appear to have carried cannabis to Europe and the Middle East. In 2016, a team from the German Archaeological Institute and the Free University, both in Berlin, found residues and botanical remains of the plant, which originates in East and Central Asia, at Yamnaya sites across Eurasia. It's difficult to know whether the Yamnaya used cannabis simply to make hemp for rope or also smoked or ingested it. But some ancient people did inhale: Digs in the Caucasus have uncovered braziers containing seeds and charred remains of cannabis dating to about 3000 B.C.E.
From here.

Wednesday, April 18, 2018

Progress Made On 68 Year Old Math Problem

How many colours are needed to colour the plane so that no two points at distance exactly 1 from each other are the same colour? 
This quantity, termed the chromatic number of the plane or CNP, was first discussed (though not in print) by Nelson in 1950 (see [Soi]). Since that year it has been known that at least four and at most seven colours are needed. 
The lower bound was also noted by Nelson (see [Soi]) and arises because there exist 4-chromatic finite graphs that can be drawn in the plane with each edge being a straight line of unit length, the smallest of which is the 7-vertex Moser spindle [MM] (see Figure 7, left panel). 
The upper bound arises because, as first observed by Isbell also in 1950 (see [Soi]), congruent regular hexagons tiling the plane can be assigned seven colours in a pattern that separates all same-coloured pairs of tiles by more than their diameter. 
The question of the chromatic number of the plane is termed the Hadwiger-Nelson problem, because of the contributions of Nelson just mentioned and because the 7-colouring of the hexagonal tiling was first discussed (though in another context) by Hadwiger in 1945 [Had]. The rich history of this problem and related ones is wonderfully documented in [Soi]. Since 1950, no improvement has been made to either bound. 
From here.

The author, a professional scientist and amateur mathematician, proves in the linked pre-print that the lower bound is 5 and not 4.

So, the correct answer is now 5, 6 or 7, although we still don't know which of those three integers is the correct answer.