The mathematical theory of big game hunting VI
Following the seminal paper by H. Petard in 1938 introducing to the mathematical theory of big game hunting, a good many others were prompted to add to this literature. The following survey artcile is one of these articles, providing an overview of key classic and new results.
We here at The komplex plane include it here, together those other articles of which we are aware. We make no claim that this is a complete compendium, but we do feel they do provide the interested reader a solid introduction into this exciting branch of mathematics.
15 new ways to catch a lion
John Barrington [1]
This, O Best Beloved, is another tale of the High and the Far-Off Times. In the blistering midst of the Sand-Swept Sahara lived a Pride of Lions. There was a Real Lion, and a Projective Lion, and a pair of Parallel Lions; and all manner of Lion Segments. And on the edge of the Sand-Swept Sahara there lived a 'nexorable Lion Hunter...
I make no apologies for raising once again the problems of the mathematical theory of big game hunting. As with any branch of mathematics, much progress has been made in the last decade.
The subject started in 1938 with the epic paper of Petard [1]. The main problem is usually formulated as follows: In the Sahara desert there exist lions. Devise methods for capturing them. Petard found ten mathematical solutions, which we can paraphrase as follows.
1. The Hilbert method. Place a locked cage in the desert. Set up the following axiomatic system.
- The set of lions is nonempty.
- If there is a lion in the desert, then there is a lion in the cage.
THEOREM 1. There is a lion in the cage.
2. The method of inversive geometry. Place a locked, spherical cage in the desert, empty of lions, and enter it. Invert with respect to the cage. This maps the lion to the interior of the cage, and you outside it.
3. The projective geometry method. The desert is a plane. Project this to a line, then project the line to a point inside the cage. The lion goes to the same point.
4. The Bolzano-Weierstrass method. Bisect the desert by a line running N-S. The lion is in one half. Bisect this half by a line running E-W. The lion is in one half. Continue this process indefinitely, at each stage building a fence. The lion is enclosed in a fence of arbitraily small length.
5. The general topology method. Observe that the desert is a separable metric space, so has a countable dense subset. Some subsequence converges to the lion. Approach stealthily along it, bearing suitable equipment.
6. The Peano method. There exists a space-filling curve passing through every point of the desert. It has been remarked [2] that such a curve may be traversed in as short a time as we please. Armed with a spear, traverse the curve faster than the lion can move his own length.
7. A topological method. The lion has at least the connectivity of a torus. Transport the desert to 4-space. It can now be deformed in such a way as to knot the lion [3]. He is now helpless.
8. The Cauchy method. Let f(z) be an analytic lion-valued function, with zeta the cage. Consider the integral
where C is the boundary of the desert. Its value is f(zeta), that is, there is a lion in the cage.
9. The Wiener Tauberian method. Procure a tame lion L0 of class L(-\infinity,\infinity) whose Fourier transform (Furrier transform?) nowhere vanishes, and set it loose in the desert. Being tame, it will converge to the cage. By Wiener [4] every other lion will converge to the same cage.
10. The Eratosthenian method. Enumerate all objects in the desert; examine them one by one; discard all those that are not lions. A refinement will capture only prime lions.
Petard also gives one physical method with strong mathematical content:
11. The Schrodinger method. At any instant there is a nonzero probability that a lion is in the cage. Wait.
The next work of any significance is that of Morhpy [5]. I confess that I do not find all of his methods convincing. The best are:
12. Surgery. The lion is an orientable 3-mainfold with boundary and so [6] may be rendered contractible by surgery. Contract him to Barnum and Bailey.
13. The cobordism method. For the same reasons the lion is a handlebody. A lion that can be handled is trivial to capture.
14. The sheaf-theoretic method. The lion is a cross-section [8] of the sheaf of the germ of lions in the desert. Re-topologize the desert to make it discrete: the stalks of the sheaf fall apart and release the germs, which kill the lion.
15. The Postnikov method. The lion, being hairy, may be regarded as a fiber space. Construct a Postnikov decomposition [9]. A decomposed lion must, of course, be long dead.
16. The universal covering. Cover the lion by his simply-connected covering space. Since this has no holes, he is trapped!
17. The game-theory method. The lion is big game, hence certainly a game. There exists an optimal strategy. Follow it.
18. The Feit-Thompson method. If necessary add a lion to make the total odd. This renders the problem soluble [10].
Recent, hitherto unpublished, work has revealed a range of new methods:
19. The field-theory method. Irrigate the desert and plant grass so that it becomes a field. A zero lion is trivial to capture, so we may assume the lion L is nonzero. The element 1 may be located just to the right of 0 in the prime subfield. Prize it appart into LL-1 and discard L-1. (Remark: the Greeks used the convention that the product of two lions is a rectangle, not a lion; the product of 3 lions is a solid, and so on. It follows that every lion is tanscendental. Modern mathematics permits algebraic lions.)
20. The kittygory method. Form the category whose objects are the lions in the desert, with trivial morphism. This is a small category (even if lions are big cats) and so can be embedded in a concrete category [11]. There is a forgetful functor from this to the category of sets: this sets the concrete and traps the embedded lions.
21. Backward induction. We prove by backward induction teh statement L(n): "It is possible to capture n lions." This is true for sufficiently large n since lions will be packed like sardines and have no room to escape. But trivially L(n+1) implies L(n), since having captured n+1 lions, we can release one. Hence L(1) is true.
22. Another topological method. Give the desert the leonine topology, in which a subset is closed if it is the whole desert, or contains no lions. The set of lions is now dense. Put an open cage in the desert. By density, it contains a lion. Shut it quickly!
23. The Moore-Smith method. Like (5) above, but this applies to non-separable deserts: the lion is caught not by a sequence, but by a net.
24. For those who insist on sequences. The real lion is non-compact and so contains non-convergent subsequences. To overcome this let Omega be the first uncountable ordinal and insert a copy of the given lion between A and A+1 for all ordinals A less than Omega. You now have a long lion in which all sequences converge [12]. Proceed as in (5).
25. The group ring method. Let G be the free group on the set L of lions, and let ZG be its group ring. The lions now belong to a ring, so are circus lions, and hence tame.
26. The Bourbaki method. The capture of a lion in the desert is a special case of a far more general problem. Formulate this problem and find necessary and sufficient conditions for its solution. The capture of a lion is now a trivial corollary of the general theory, which on no account should be written down explicitly.
27. The Hasse-Minkowski method. Consider the lion-catching problem modulo p for all primes p. There being only finitely many possibilities, this can be solved. Hence, the original problem can be solve [13].
28. The PL method. The lion is a 3-manifold with nonempty boundary. Triangulate it to get a PL manifold. This can be collared [14], which is what we wish to achieve.
29. The singularity method. Consider a lion in the plane. If it is a regular lion its regular habits render it easy to catch (e.g. dig a pit). Without loss of generality, it is a singular lion. Stable singularities are dense, so without loss of generality the lion is stable. The singularity is not a self-intersection (since a self-intersecting lion is absurd) so it must be a cusp. Complexify and intersect with a sphere to get a trefoil knot. As in (7) the problem becomes trivial.
30. The measure-theoretic method. Assume for a contradiction that no lion can be captured. Since capturable lions are imaginary, all lions are real. On any real lion there exists a nontrivial invariant measure m, namely Harr or Lebesgue measure. Then the product m x m is a Baire measure on L x L by [15]. Since a product of lions cannot be a bear, the Baire measure on L x L is zero. Hence, m=0, a contradiction. Thus, all lions may be captured.
31. The method of parallels. Select a point in the desert and introduce a tame lion not passing through that point. There are three cases.
- The geometry is Euclidean. There is then a unique parallel lion passing through the selected point. Grab it as it passes.
- The geometry is hyperbolic. The same method will now catch infinitely many lions.
- The geometry is elliptic. There are no parallel lions, so every lion meets every other lion. Follow the tame lion and catch all the lions it meets: in this way, every lion in the desert will be captured.
32. The Thom-Zeeman method. A lion loose in the desert is an obvious catastrophe [16]. It has three dimensions of control (2 for position, 1 for time) and one dimension of behavior (being parametrized by a lion). Hence by Thom's Classification Theorem it is a swallowtail. A lion that has swallowed its tail is in no state to avoid capture.
33. The Australian method. Lions are very varied creatures, so there is a variety of lions in the desert. This variety contains free lions [17], which satisfy no nontricial identities. Select a lion and register it as "Fred Lion" at the local register Office: now it has a nontrivial identity, and hence cannot be free. If it is not free it must be captive. (If "Fred Lion" is thought to be a trivial identity, call it "Albert Einstein.")
Bibliogoraphy
[1] H. Petard. A contribution to the mathematical theory of big game hunting, Amer. Math. Monthly 45 (1938) 446-7.
[2] E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series, 1927.
[3] H. Seifert and W. Threlfall, Lehrbuch der Topologie, 1934.
[4] N. Wiener, The Fourier integral and certain of its applications, 1933.
[5] O. Morphy, Some modern mathematical methods in the theory of big game hunting, Amer. Math. Monthly 75 (1968) 185-7.
[6] M. Kervaire and J. Milnor, Groups of homotopy spheres I, Ann. of Math. 1963.
[7] This footnote has been censored by the authorities.
[8] It has been verified experimentally that lions are cross.
[9] E. Spanier, Algebraic Topology, McGraw-Hill 1966.
[10] W. Feit and J. G. Thompson, Solvability of groups of odd order, Pac. J. Math. 1963.
[11] P. Freyd, Abelian Categories.
[12] J. L. Kelley, General Topology.
[13] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, 1973.
[14] C. P. Rourke and B. L. Sanderson, Itroduction to Piecewise Linear Topology, 1973.
[15] S. K. Berberian, Topological Groups.
[16] R. Thom, Stabilite Structurelle et Morphogenese, 1972.
[17] Hanna Neumann, Varieties of Groups, 1972.