Original Math Notes
All Seasons
Episode 412: Power--Interactive Computations--Explore the Math
Episode 412: Power
Exploring Venn-like configurations by inversion of a chain of circles with centers falling on a congruent circle centered at the origin
Scene 3:
INT. CALSCI - CHARLIE'S OFFICE - DAY

Combinatorial matrix theory (again)

AMITA packs her bag for class as CHARLIE writes a list on the
board-- Non-negative matrices, linear programming, symmetric
matrices, etc.

                   CHARLIE
          Shall we call it 'Matrix Theory
          with Combinatorics?'

                   AMITA
          How about 'Combinatorial
          Applications?'  We should teach
          together-- you explain the theory,
          I show the applications.

                   CHARLIE
          Great, when are you free to plan
          presentations?

                   AMITA
              (ready to leave)
          I'm teaching until four, then I
          have office hours until six.  After
          that, over dinner?

                   CHARLIE
          Sure.

As we know from Episode 407, "Primacy," Amita's teaching and research interests include combinatorial matrix theory. This is a rich branch of mathematics that combines combinatorics, graph theory, and linear algebra.

Nonnegative matrices

Nonnegative matrices are important in a variety of applications and have a number of attractive mathematical properties. Together with positive semidefinite matrices, they serve as as a natural generalization of nonnegative real numbers. The most fundamental properties of nonnegative matrices require fairly advanced mathematics and were established by Oskar Perron in 1907 and Ferdinand Georg Frobenius 1912. To quote Charles Johnson in his review of the classic text on the subject by Abraham Berman and Robert J. Plemmons, "Important properties [of nonnegative matrices] are still being discovered, and, typical of much of modern research in matrix theory, the work often involves an attractive marriage of algebra, analysis, combinatorics, and geometry."

Linear programming

Linear programming refers to optimizing an outcome using a linear mathematical model together with a set of constraints. In the Season 4 opening episode "Trust Metric," Charlie used set covering deployment (a particular application of a specialized type of linear programming) to help place police units optimally. In the same episode, he also remarked, "You don't need Karmarkar's algorithm" to mean "You don't need to be a rocket scientist to know...." In that quote, Karmarkar's algorithm refers to a so-called interior point method of linear programming that is generally much more efficient than traditional techniques (such as the simplex method).

Symmetric (and antisymmetric) matrices

A symmetric matrix is a square (and generally real) matrix whose transpose (i.e., the matrix obtained by interchanging rows and columns) equals itself. An antisymmetric matrix is similarly defined as a matrix whose transpose is the negative of itself. Symmetric matrices are especially important in mathematics and physics because the generalization of symmetric matrices to the case of matrices having complex-valued elements are objects known as Hermitian matrices, which are fundamental in the theory of quantum mechanics and its relativistic generalizations.

Matrices may be decomposed into their symmetric and antisymmetric parts via

Matrix decomposition into symmetric and antisymmetric parts

where

Matrix A and Matrix A-transpose

so

(2 times the) symmetric part of matrix A

(twice the symmetric part), which is symmetric, and

(2 times the) antisymmetric part of matrix A

(twice the antisymmetric part), which is antisymmetric, and half the sum of these gives the original matrix.

Scene 6:
INT. CALSCI - CHARLIE'S OFFICE - DAY

Fish bladders, lenses, and intersecting circles

                   LARRY
              (re: the board)
          Do you know what this is?

                   CHARLIE
          A Venn Diagram Intersection
          indicating suspects who match the
          description and have a history of
          violence.  I'm helping Don.

                   LARRY
          It's also an ancient religious
          symbol-- you may recognize it--
              (outlines the "fish")
          The Vesica Piscis, symbolizing the
          meeting of heaven and earth.

Megan has entered, seeing what Larry's drawn.

                   MEGAN
          You know the coolest things--

Lenses

Lens formed by two intersecting circles

In mathematics, a lens is a convex plane figure composed of two circular arcs. The term vesica piscis (Latin for fish bladder) is often used to refer to the particular lens formed by the intersection of two unit circles whose centers are offset by a unit distance--a configuration illustrated above. Here, the lens is indicated as the two dark black curves enclosing the shaded interior region. The symmetric lens obtained in this way has been endowed with various mystical properties through the ages, apparently first among the Pythagoreans, who considered it a holy figure. Renaissance artists also frequently surrounded images of Jesus with the vesica piscis.

The height-to-width-ratio of the vesica piscis is equal to sqrt(3), the square root of 3, a number sometimes known as Theodorus's constant. Compare this to sqrt(2), the square root of 2, which is sometimes known as Pythagoras's constant and was of even more significance to the Pythagoreans. In fact, legend holds that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of sqrt(2) (i.e., the fact that this number cannot be expressed as the ratio of any two integers) while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans.

Formula for the region of intersection of two circles

Note that we encountered circle intersections previously in Episode 401, "Trust Metric", as well as in Episode 409, "Graphic." The area of a lens, corresponding to the area of overlap indicated in aqua above, can be determined directly from the trigonometric formula given in the Episode 409 notes and reproduced above by setting d = r = R = 1. Alternately, it can derived with the help of a little integral calculus, which is especially easy using Mathematica's ability to integrate over regions specified by inequalities (this is done internally using cylindrical algebraic decomposition) as follows:

Integral for the area of a lens

In the above piece of code, the two inequalities are simply those specifying the interiors of the two circles whose intersection gives the lens. This produces the beautiful result:

Lens area

Lunes

Lunes

Geometric figures related to lenses that consist of crescent-shaped (i.e., concave) regions bounded by two circular arcs of unequal radii were also considered by the ancient Greeks and are known as lunes. In each of the figures above, the area of the lune (filled in blue) is equal to the area of the indicated triangle (filled in yellow). Hippocrates of Chios squared the above left lune as well as two others in the 5th century BC. Two more squarable lunes were found by T. Clausen in the 19th century. Not until the 20th century did N. G. Tschebatorew and A. W. Dorodnow finally prove that these are the only five squarable lunes.

Mystical circle arrangements

There are many attractive arrangements of circles that give symmetrically placed lens-shaped regions, including a number found at the Temple of Osiris at Abydos, Egypt. Configurations sometimes known as the seed of life (which also appeared in Italian art from the 13th century) and the flower of life (which appeared in Phoenician art from the 9th century BC) are illustrated below.

Seed of life (left); flower of life (right)
Scene 6:
INT. CALSCI - CHARLIE'S OFFICE - DAY

Venn diagrams

                   CHARLIE
          Venn diagrams are illustrations
          used in set theory, showing
          relationships among groups of
          things.  We've deepened the
          analysis from what you learned in
          school by adding dimensions.
Venn diagrams

A Venn diagram is a schematic diagram used in logic theory to depict collections of sets and represent their relationships. The Venn diagrams on two and three sets are illustrated above. The order-two diagram (left) consists of two intersecting circles, producing a total of four regions, A, B, AB, and the empty set ∅, represented by none of the regions occupied. Here, AB denotes the intersection of sets A and B. The order-three diagram (right) consists of three symmetrically placed mutually intersecting circles comprising a total of eight regions. The regions labeled A, B, and C consist of members that are only in one set and no others; the three regions labeled AB, AC, and BC consist of members that are in two sets but not the third; the region ABC consists of members that are simultaneously in all three; and no regions occupied represents ∅.

In general, an order-n Venn diagram is a collection of n simple closed curves in the plane with the following properties:

  1. The curves partition the plane into 2n connected regions.
  2. Each subset S of {1, 2, ..., n} corresponds to a unique region formed by the intersection of the interiors of the curves in S.
Symmetric Venn diagrams of orders 5, 7, and 11. Reproduced from Ruskey, Savage, and Wagon (2006).
Symmetric Venn diagrams of orders 5, 7, and 11

The left figure above is an n = 5 Venn diagram by Branko Grünbaum, while the attractive 7-fold rosette illustrated in the middle figure is an n = 7 Venn diagram, called "Victoria" by Frank Ruskey. The right figure shows a recently constructed symmetric Venn diagram on n = 11, due to Ruskey, Carla Savage, and Stan Wagon. See the papers and online material in the references section below for many more details and attractive figures, as well as for interesting recent developments in the field of Venn diagram research.

Scene 15:
INT. EPPES HOUSE - LIVING ROOM - NIGHT

The Higgs boson

                   LARRY
          Dr. Ramanujan, I have a proposition
          for you.

                   AMITA
              (teasing)
          Better not tell Charlie...

                   LARRY
          Once or twice in a lifetime there
          are quests that require all we have
          to give and more.  I am embarking
          on such a quixotic adventure and in
          thinking on who should be my Sancho
          Panza I come up with only one name--
          yours.

                   AMITA
          Larry, what are you talking about?

                   LARRY
          I've accepted the D Zero team's
          offer to search for the Higgs Boson
          and I require a computational
          partner.
Animation of Higgs emission

As already discussed in Episode 407, "Primacy," one of the main unconfirmed aspects of the "standard model" that unites three of the four known forces of the universe is the existence (and the properties) of the so-called Higgs boson. This particle is a very important theoretical ingredient of the theory, so finding it in Nature is important to ensure the full correctness of the standard model. A schematic animation of Higgs emission is shown at right.


References

Berman, A. and Plemmons, R. J. Nonnegative Matrices in the Mathematical Sciences. SIAM, 1994.

Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 1-20, 1990.

Frobenius, G. "Über Matrizen aus nicht negativen Elementen." S.-B. Preuss. Akad. Wiss. (Berlin), pp. 456--477, 1912.

Grünbaum, B. "On Venn Diagrams and the Counting of Regions." College Math. J. 15, 433-435, 1984.

Grünbaum, B. "Venn Diagrams and Independent Families of Sets." Math. Mag. 48, 12-23, 1975.

Johnson, C. R. Book review. "Nonnegative Matrices in the Mathematical Sciences, by Abraham Berman and Robert J. Plemmons." Bull. Amer. Math. Soc. 6, 233-235, 1981.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., p. xii, 1995.

Perron, O. "Zur theorie der matrizen." Math. Ann. 64, 248-263, 1907.

Ruskey, F. "A Survey of Venn Diagrams." Electronic J. Combinatorics, Dynamical Survey DS5, June 18, 2005. http://www.combinatorics.org/Surveys/#DS5.

Ruskey, F. "Venn Diagrams." http://www.combinatorics.org/Surveys/ds5/VennEJC.html.

Ruskey, F.; Savage, C. D.; and Wagon, S. "The Search for Simple Symmetric Venn Diagrams." Notices Amer. Math. Soc. 53, 1304-1311, 2006. PDF

Shenitzer, A. and Steprans, J. "The Evolution of Integration." Amer. Math. Monthly 101, 66-72, 1994.

Venn, J. "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings." Dublin Philos. Mag. J. Sci. 9, 1-18, 1880.

Wikipedia. "Vesica_piscis."

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 43 and 873, 2002.

 
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