Searching (again!?) for the SS Central America
On Tuesday, September 8th 1857, the steamboat SS Central America left Havana at 9 AM for New York, carrying about 600 passengers and crew members. Inside of this vessel, there was stowed a very precious cargo: a set of manuscripts by John James Audubon, and three tons of gold bars and coins. The manuscripts documented an expedition through the yet uncharted southwestern United States and California, and contained 200 sketches and paintings of its wildlife. The gold, fruit of many years of prospecting and mining during the California Gold Rush, was meant to start anew the lives of many of the passengers aboard.
On the 9th, the vessel ran into a storm which developed into a hurricane. The steamboat endured four hard days at sea, and by Saturday morning the ship was doomed. The captain arranged to have women and children taken off to the brig Marine, which offered them assistance at about noon. In spite of the efforts of the remaining crew and passengers to save the ship, the inevitable happened at about 8 PM that same day. The wreck claimed the lives of 425 men, and carried the valuable cargo to the bottom of the sea.
It was not until late 1980s that technology allowed recovery of shipwrecks at deep sea. But no technology would be of any help without an accurate location of the site. In the following paragraphs we would like to illustrate the power of the scipy stack by performing a simple simulation, that ultimately creates a dataset of possible locations for the wreck of the SS Central America, and mines the data to attempt to pinpoint the most probable target.
We simulate several possible paths of the steamboat (say 10,000 randomly generated possibilities), between 7:00 AM on Saturday, and 13 hours later, at 8:00 pm on Sunday. At 7:00 AM on that Saturday the ship’s captain, William Herndon, took a celestial fix and verbally relayed the position to the schooner El Dorado. The fix was 31º25′ North, 77º10′ West. Because the ship was not operative at that point—no engine, no sails—, for the next thirteen hours its course was solely subjected to the effect of ocean current and winds. With enough information, it is possible to model the drift and leeway on different possible paths.
Book presentation at the USC Python Users Group
More on Lindenmayer Systems
We briefly explored Lindenmayer systems (or L-systems) in an old post: Toying with Basic Fractals. We quickly reviewed this method for creation of an approximation to fractals, and displayed an example (the Koch snowflake) based on tikz libraries.
I would like to show a few more examples of beautiful curves generated with this technique, together with their generating axiom, rules and parameters. Feel free to click on each of the images below to download a larger version.
Note that any coding language with plotting capabilities should be able to tackle this project. I used once again tikz for , but this time with the tikzlibrary lindenmayersystems.
Would you like to experiment a little with axioms, rules and parameters, and obtain some new pleasant curves with this method? If the mathematical properties of the fractal that they approximate are interesting enough, I bet you could attach your name to them. Like the astronomer that finds through her telescope a new object in the sky, or the zoologist that discover a new species of spider in the forest.
Some results related to the Feuerbach Point
Given a triangle the circle that goes through the midpoints of each side, is called the Feuerbach circle. It has very surprising properties:
- It also goes through the feet of the heights, points
- If denotes the orthocenter of the triangle, then the Feuerbach circle also goes through the midpoints of the segments For this reason, the Feuerbach circle is also called the nine-point circle.
- The center of the Feuerbach circle is the midpoint between the orthocenter and circumcenter of the triangle.
- The area of the circumcircle is precisely four times the area of the Feuerbach circle.
Most of these results are easily shown with sympy without the need to resort to Gröbner bases or Ritt-Wu techniques. As usual, we realize that the properties are independent of rotation, translation or dilation, and so we may assume that the vertices of the triangle are and for some positive parameters To prove the last statement, for instance we may issue the following:
>>> import sympy >>> from sympy import * >>> A=Point(0,0) >>> B=Point(1,0) >>> r,s=var('r,s') >>> C=Point(r,s) >>> D=Segment(A,B).midpoint >>> E=Segment(B,C).midpoint >>> F=Segment(A,C).midpoint >>> simplify(Triangle(A,B,C).circumcircle.area/Triangle(D,E,F).circumcircle.area) 4
But probably the most amazing property of the nine-point circle, is the fact that it is tangent to the incircle of the triangle. With exception of the case of equilateral triangles, both circles intersect only at one point: the so-called Feuerbach point.
An Automatic Geometric Proof
We are familiar with that result that states that, on any given triangle, the circumcenter, centroid and orthocenter are always collinear. I would like to illustrate how to use Gröbner bases theory to prove that the incenter also belongs in the previous line, provided the triangle is isosceles.
We start, as usual, indicating that this property is independent of shifts, rotations or dilations, and therefore we may assume that the isosceles triangle has one vertex at , another vertex at and the third vertex at for some value In that case, we will need to work on the polynomial ring since we need the parameter free, the variables and are used to input the conditions for the circumcenter of the triangle, the variables and for centroid, and the variables and for the incenter (note that we do not need to use the orthocenter in this case).
We may obtain all six conditions by using sympy, as follows:
>>> import sympy >>> from sympy import * >>> A=Point(0,0) >>> B=Point(1,0) >>> s=symbols("s",real=True,positive=True) >>> C=Point(1/2.,s) >>> T=Triangle(A,B,C) >>> T.circumcenter Point(1/2, (4*s**2 - 1)/(8*s)) >>> T.centroid Point(1/2, s/3) >>> T.incenter Point(1/2, s/(sqrt(4*s**2 + 1) + 1))
This translates into the following polynomials
(for centroid)
(for incenter)
The hypothesis polynomial comes simply from asking whether the slope of the line through two of those centers is the same as the slope of the line through another choice of two centers; we could use then, for example, It only remains to compute the Gröbner basis of the ideal Let us use SageMath for this task:
sage: R.<s,x1,x2,x3,y1,y2,y3,z>=PolynomialRing(QQ,8,order='lex') sage: h=[2*x1-1,8*r*y1-4*r**2+1,2*x2-1,3*y2-r,2*x3-1,(4*r*y3+1)**2-4*r**2-1] sage: g=(x2-x1)*(y3-y1)-(x3-x1)*(y2-y1) sage: I=R.ideal(1-z*g,*h) sage: I.groebner_basis() [1]
This proves the result.