John Edmark is an artist and professor at Stanford University who has used the Golden Angle to sculpt spirals. The Golden Angle is derived from the Golden Ratio: it is the smaller of the two angles created by dividing the circumference of a circle according to the golden ratio and comes out to be around 137.5°.
Today I thought I’d share a video that I came across the other day. Hope you enjoy!
“Bacteria and viruses hitch a ride inside droplets of all kinds—sneezes, raindrops, toilet splatter. By reviewing footage of different types of drops, applied mathematician Lydia Bourouiba records and measures where they disperse in order to better understand how diseases spread. Watch how Bourouiba designs tests—some inescapably humorous and awkward—to study infectious disease transmission.”
The Kakeya needle problem asks whether there is a minimum area of a region in the plane in which a line segment of width 1 can be freely rotated through 360°, where translation of the segment is allowed.
This question was first posed for convex regions in 1917 by mathematician Sōichi Kakeya. It was shown by Gyula Pál that the minimum area for convex regions is achieved by an equilateral triangle of height 1 and area 1/√3.
Kakeya suggested that the minimum area, without the convexity restriction, would be a three-pointed deltoid shape. However, this is false.
Needle rotating inside a deltoid | Source: Wikipedia
Besicovitch was able to show that there is no lower bound >0 for the area of a region in which a needle of unit length can be turned around. The proof of this relies on the construction of what is now known as a Besicovitch set, which is a set of measure zero in the plane which contains a unit line segment in every direction.
One can construct a set in which a unit line segment can be rotated continuously through 180 degrees from a Besicovitch set consisting of Perron trees.
Kakeya Needle Set constructed from Perron trees | Source: Wikipedia
However, although there are Kakeya needle sets of arbitrarily small positive measure and Besicovich sets of measure 0, there are no Kakeya needle sets of measure 0.
Today I wanted to share with you a video which I came across the other day on the Map of Mathematics.
Although many people view maths as synonymous with pain, boredom or frustration, one must appreciate its diversity and vast implications on other subjects; you may not have the background to see beauty in a particular equation, but virtually anyone can appreciate the amazing advancements humans have made from basic counting to creating full-on artificial intelligence.
“While an artistic temperament is often considered the exact opposite of the kind of personality that loves complicated equations, pure mathematicians are really just a bunch of lunatics endlessly working with abstraction and beauty.“
– Rhett Jones
In the video below, Dominic Walliman takes viewers through the major fields of math starting at the beginning and shows us how they inform and relate to each other. Of course many details have been left out, as to properly connect the various disciplines of math we would need a 3D web! Also, in reality, “the study of math’s foundations has yet to discover a complete and consistent set of axioms.“
Exponents, logarithms and roots each utilise different notation, which can be confusing for students as it’s almost like learning three different ‘languages’, for no real good reason. The video I am sharing with you, by 3blue1brown, attempts to simplify this notation by using one elegant system, and also explains how maths could be made more accessible by developing cleaner notation.