A contribution to the mathematical theory of big game hunting
H. Pétard Princeton, New Jersey
This little known mathematical discipline has not, of recent years, received in the literature the attention which, in our opinion, it deserves. In the present paper we present some algorithms which, it is hoped, may be of interest to other workers in the field. Neglecting the more obviously trivial methods, we shall confine our attention to those which involve significant applications of ideas familiar to mathematicians and physicists.
The present time is particularly fitting for the preparation of an account of the subject, since recent advances both in pure mathematics and in theoretical physics have made available powerful tools whose very existence was unsuspected by earlier investigators. At the same time, some of the more elegant classical methods acquire new significance in the light of modern discoveries. Like many other branches of knowledge to which mathematical techniques have been applied in recent years, the Mathematical Theory of Big Game Hunting has a singularly happy unifying effect of the most diverse branches of the exact sciences.
For the sake of simplicity of statement, we shall confine our attention to lions (Felis leo) whose habitat is the Sahara Desert. The methods which we shall enumerate will easily be seen to be applicable, with obvious formal modifications, to other carnivores and to other portions of the globe. The paper is divided into three parts, which draw their material respectively from mathematics, theoretical physics, and experimental physics.
The author desires to acknowledge his indebtness to the Trivial Club, St. John's College, Cambridge, England; to the MIT chapter of the Society for Useless Research; to the F o P, of Princeton University; and to numerous individual contributors, known and unknown, conscious and unconscious.
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1. Mathematical Methods
1.1 The Hilbert, or axiomatic, method
We place a locked cage at a given point of the desert. We then introduce the following logical system.
Axiom 1: The class of lions in the Sahara Desert is non-void.
Axiom 2: If there is a lion in the Sahara Desert, there is a lion in the cage.
Rule of procedure: If p is a theorem, and 'p implies q' is a theorem, then q is a theorem.
Theorem 1: There exists a lion in the cage.
1.2 The method of inversive geometry
We place a spherical cage in the desert, enter it and lock it. We perform an inversion with respect to the cage. The lion is then in the interior of the cage, and we are outside.
1.3 The method of projective geometry
Without loss of generality, we may regard the Sahara Desert as a plane. Project the plane into a line, and then project the line into an interior point of the cage. The lion is projected into the same point.
1.4 The Bolzano-Weierstrass method
Bisect the desert by a line running N-S. The lion is either in the E portion or in the W portion; let us suppose him to be in the W portion. Bisect this portion by a line running E-W. The lion is either in the N portion or in the S portion; let us suppose him to be in the N portion. We continue this process indefinitely, constructing a sufficiently strong fence about the chosen portion at each step. The diameter of the chosen portions approaches zero, so that the lion is ultimately surrounded by a fence of arbitrarily small diameter.
1.5 The 'Mengentheoretisch' [set-theoretical] method
We observe that the desert is a separable space. It therefore contains an enumerable dense set of points, from which can be extracted a sequence having the lion as limit. We then approach the lion stealthily along this sequence, bearing with us suitable equipment.
1.6 The Peano method
Construct, by standard methods, a continuous curve passing through every point of the desert. It has been remarked1 that it is possible to traverse such a curve in an arbitrarily short time. Armed with a spear, we traverse the curve in a time shorter than that in which a lion can move his own length.
1.7 A topological method
We observe that the lion has at least the connectivity of the torus. We transport the desert into four-space. It is then possible2 to carry out such a deformation that the lion can be returned to three-space in a knotted condition. He is then helpless.
1.8 The Cauchy, or function-theoretical, method
We consider an analytic lion-valued function f(z). Let zeta be the cage. Consider the integral where C is the boundary of the desert; its value is f(zeta), i.e., a lion in the cage.3
1.9 The Wiener Tauberian method
We procure a tame lion, L0 of class L(-00, 00), whose Fourier transform nowhere vanishes, and release it in the desert. L0 then converges to our cage. By Wiener's General Tauberian theorem4, any other lion, L (say), will then converge to the same cage. Alternatively, we can approximate arbitrarily closely to L by translating L0 about the desert5.
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2. Methods from Theoretical Physics
2.1 The Dirac method
We observe that wild lions are, ipso facto, not observable in the Sahara Desert. Consequently, if there are any lions in the Sahara, they are tame. The capture of a tame lion may be left as an exercise for the reader.
2.2 The Schrödinger method
At any given moment there is a positive probability that there is a lion in the cage. Sit down and wait.
2.3 The method of nuclear physics
Place a tame lion in the cage, and apply a Majorana exchange operator6 between it and a wild lion.
As a variant, let us suppose, to fix ideas, that we require a male lion. We place a tame lioness in the cage, and apply a Heisenberg exchange operator7, which exchanges the spins.
2.4 A relativistic method
We distribute about the desert lion bait containing large portions of the Companion of Sirius. When enough bait has been taken, we project a beam of light across the desert. This will bend right round the lion, who will then become so dizzy that he can be approached with impunity.
2.5 The Newton method
Neglect friction and the lion and the cage will attract each other.
2.6 The Heisenberg method
You will disturb the lion if you observe it before its capture, so keep your eyes closed.
2.7 The Einstein method
Run in the direction opposite to that of the lion. The relative velocity makes the lion run faster, and hence it feels heavier.
2.8 The quantum measurement method
We assume that the sex of the lion is ab initio indeterminate. The wave function for the lion is hence a superposition of the gender eigenstate for a lion and that for a lioness. We lay these eigenstates out flat on the ground and orthogonal to each other. Since the (male) lion has a distinctive mane, the measurement of sex can safely be made from a distance, using binoculars. The lion then collapses into one of the eigenstates, which is rolled up and placed inside the cage.
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3. Methods from Experimental Physics
3.1 The thermodynamical method
We construct a semi-permeable membrane, permeable to everything except lions, and sweep it across the desert.
3.2 The atom-splitting method
We irradiate the desert with slow neutrons. The lion becomes radioactive, and a process of disintegration sets in. When the decay has proceeded sufficiently far, he will become incapable of showing fight.
3.3 The magneto-optical method
We plant a large lens-shaped bed of catnip (Nepata cataria) whose axis lies along the direction of the horizontal component of the earth's magnetic field, and place a cage at one of the field's foci. We distribute over the desert large quantities of magnetized spinach (Spinacia oleracea), which, as is well known, has a high ferric content. The spinach is eaten by the herbivorous denizens of the desert, which are in turn eaten by lions. The lions are then oriented parallel to the earth's magnetic field, and the resulting beam of lions is focused by the catnip upon the cage.
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4. Contributions from Computer Science
4.1 The search method
We assume that the lion is most likely to be found in the direction to the north of the point where we are standing. Therefore the real problem we have is that of speed, since we are only using a PC to solve the problem.
4.2 The parallel search method
By using parallelism, we will be able to search in the direction to the north much faster than earlier.
4.3 The Monte Carlo method
We pick a random number indexing the space we search. By excluding neighbouring points in the search, we can drastically reduce the number of points we need to consider. The lion will according to probability appear sooner or later.
4.4 The practical approach
We see a rabbit very close to us. Since it is already dead, it is particularly easy to catch. We therefore catch it and call it a lion.
4.5 The common language approach
If only everyone used ADA/Common Lisp/Prolog, this problem would be trivial to solve.
4.6 The standard approach
We know what a Lion is from ISO 4711/X.123. Since CCITT have specified a Lion to be a particular option of a cat, we will have to wait for a harmonised standard to appear. Funding worth 10,000,000 pounds has been provided for initial investigations into this standard's development.
4.7 Linear search
Stand in the top left-hand corner of the Sahara Desert. Take one step east. Repeat until you have found the lion or you reach the right-hand edge. If you reach the right-hand edge, take one step southwards, and proceed towards the left-hand edge. When you finally reach the lion, put it in the cage. If the lion should happen to eat you before you manage to get it in the cage, press the reset button, and try again.
4.8 The Dijkstra approach
The way the problem reached us was: 'Catch a wild lion in the Sahara Desert'. Another way of stating the problem is:
Axiom 1: Sahara elem deserts
Axiom 2: Lion elem Sahara
Axiom 3: NOT(Lion elem cage)
We observe the following invariant:
P1: C(L) v not(C(L))
where C(L) means: the value of L is in the cage.
Establishing C initially is trivially accomplished with the statement
;cage := {}
Note 0:
This is easily implemented by opening the door to the cage and shaking out any lions that happen to be there initially.
(End of note 0.)
The obvious program structure is then:
;cage:={}
;do NOT (C(L)) ->
;'approach lion under invariance of P1'
;if P(L) ->
;'insert lion in cage'
[] not P(L) ->
;skip
;fi
;od
where P(L) means: the value of L is within arm's reach.
Note 1:
Axiom 2 ensures that the loop terminates.
(End of note 1.)
Exercise 0:
Refine the step 'Approach lion under invariance of P1'.
(End of exercise 0.)
Note 2:
The program is robust in the sense that it will lead to abortion if the value of L is 'lioness'.
(End of note 2.)
Remark 0:
This may be a new sense of the word 'robust' for you.
(End of remark 0.)
Note 3:
From observation we can see that the above program leads to the desired goal. It goes without saying that we therefore do not have to run it.
(End of note 3.)
(End of approach.)
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The following paper was published in the American Mathematical Monthly 72 (1965) p. 436.
5. A New Method of Catching a Lion
I. J. Good
In this note a definitive procedure will be provided for catching a lion in a desert.
Let Q be the operator that encloses a word in quotation marks. Its square Q^2 encloses a word in double quotes. The operator clearly satisfies the law of indices, Q^m*Q^n=Q^(m+n). Write down the word 'lion', without quotation marks. Apply to it the operator Q^(-1). Then a lion will appear on the page. It is advisable to enclose the page in a cage before applying the operator.
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The following paper was published in the American Mathematical Monthly 74 (1967) pp. 838-839.
6. On a Theorem of H. Pétard
Christian Roselius Tulane University
In a classical paper, H. Pétard proved that it is possible to capture a lion in the Sahara desert. He further showed3 that it is in fact possible to catch every lion with at most one exception. Using completely new techniques, not available to Pétard at the time, we are able to sharpen this result, and to show that every lion may be captured.
Let L denote the category whose objects are lions, with 'ancestor' as the only nontrivial morphism. Let l be the category of caged lions. The subcategory l is clearly complete, is nonempty (by inspection), and has both generator and cogenerator8. Let F:l->L be the forgetful functor, which forgets the cage. By the Adjoint Functor Theorem9, 80-91, the functor F has a coadjoint C:L->l, which reflects each lion into a cage.
We remark that this method is obviously superior to the Good method, which only guarantees the capture of one lion, and which requires an application of the Weierkäfig Preparation Theorem.
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The following paper was published in the American Mathematical Monthly 75 (1968) pp. 185-187.
7. Some Modern Mathematical Methods in the Theory of Lion Hunting10
Otto Morphy, D.Hp. (Dr. of Hypocrisy)
It is now 30 years since the appearance of H. Pétard's classic treatise on the mathematical theory of big game hunting. These years have seen a remarkable development of practical mathematical techniques. It is, of course, generally known that it was Pétard's famous letter to the president in 1941 that led to the establishment of the Martini Project, the legendary crash programme to develop new and more efficient methods for search and destroy operations against the Axis lions. The Infernal Bureaucratic Federation (IBF) has recently declassified certain portions of the formerly top secret Martini Project work. Thus we are now able to reveal to the world, for the first time, these important new applications of modern mathematics to the theory and practice of lion hunting. As has become standard practice in the discipline we shall restrict our attention to the case of lions residing in the Sahara Desert11. As noted by Pétard, most methods apply, more generally, to other big game. However, method (3) below appears to be restricted to the genus Felis. Clearly, more research on this important matter is called for.
7.1 Surgical method
A lion may be regarded as an orientable three-manifold with a nonempty boundary. It is known12 that by means of a sequence of surgical operations (known as 'spherical modifications' in medical parlance) the lion can be rendered contractible. He may then be signed to a contract with Barnum and Bailey.
7.2 Logical method
A lion is a continuum. According to Cohen's theorem13 he is undecidable (especially when he must make choices). Let two men approach him simultaneously. The lion, unable to decide upon which man to attack, is then easily captured.
7.3 Functorial method
A lion is not dangerous unless he is somewhat gory. Thus the lion is a category. If he is a small category then he is a kittygory9 and certainly not to be feared. Thus we may assume, without loss of generality, that he is a proper class. But then he is not a member of the universe and is certainly not of any concern to us.
7.4 Method of differential topology
The lion is a three-manifold embedded in Euclidean 3-space. This implies that he is a handlebody14. However, a lion which can be handled is tame and will enter the cage upon request.
7.5 Sheaf theoretic method
The lion is a cross-section15 of the sheaf of germs of lions16 on the Sahara Desert. Merely alter the topology of the Sahara, making it discrete. The stalks of the sheaf will then fall apart releasing the germs which attack the lion and kill it.
7.6 Method of transformation groups
Regard the lion as a surface. Represent each point of the lion as a coset of the group of homeomorphisms of the lion modulo the isotropy group of the nose (considered as a point)17. This represents the lion as a homogeneous space. That is, this representation homogenizes the lion. A homogenized lion is in no shape to put up a fight18.
7.7 Postlikov method
A male lion is quite hairy19 and may be regarded as being made up of fibres. Thus we may regard the lion as a fibre space. We may then construct a Postlikov decomposition20 of the lion. This being done, the lion, being decomposed, is dead and in bad need of burial.
7.8 Steenrod algebra method
Consider the mod p cohomology ring of the lion. We may regard this as a module over the mod p Steenrod algebra. Doing this requires the use of the table of Steenrod cohomology operations21. Every element must be killed by some of these operations. Thus the lion will die on the operating table.
7.9 Homotopy method
The lion has the homotopy type of a one-dimensional complex and hence he is a K(Pi, 1) space. If Pi is noncommutative then the lion is not a member of the international commutist conspiracy22 and hence he must be friendly. If Pi is commutative then the lion has the homotopy type of the space of loops on a K(Pi, 2) space20. We hire a stunt pilot to loop the loops, thereby hopelessly entangling the lion and rendering him helpless.
7.10 Covering space method
Cover the lion by his simply connected covering space. In effect this decks the lion23. Grab him while he is down.
7.11 Game theoretic method
A lion is big game. Thus, a fortiori, he is a game. Therefore there exists an optimal strategy24. Follow it.
7.12 Group theoretic method
If there are an even number of lions in the Sahara Desert we add a tame lion. Thus we may assume that the group of Sahara lions is of odd order. This renders the situation capable of solution according to the work of Thompson and Feit25.
We conclude with one significant nonmathematical method:
7.13 Biological method
Obtain a number of planarians and subject them to repeated recorded statements saying: 'You are a planarian'. The worms should shortly learn this fact since they must have some suspicions to this effect to start with. Now feed the worms to the lion in question. The knowledge of the planarians is then transferred to the lion26. The lion, now thinking that he is a planarian, will proceed to subdivide. This process, while natural for the planarian, is disastrous to the lion27.
Ed. note: Prof. Morphy is the namesake of his renowned aunt, the author of the famous series of epigrams now popularly known as Auntie Otto Morphisms or euphemistically as epimorphisms.
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The following paper was published in the American Mathematical Monthly 75 (1968) pp. 896-897.
8. Further Techniques in the Theory of Big Game Hunting
Patricia L. Dudley, G. T. Evans, K. D. Hansen and I. D. Richardson Carleton University, Ottawa
Interest in the problem of big game hunting has recently been reawakened by Morphy's paper in this MONTHLY, Feb. 1968, p. 185. We outline below several new techniques, including one from the humanities. We are also in possession of a solution by means of Bachmann geometry which we shall be glad to communicate to anyone who is interested.
8.1 Moore-Smith method
Letting A = Sahara Desert, one can construct a net in A converging to any point in the closure of A. Now lions are unable to resist tuna fish, on account of the charge atoms found therein (see Galileo Galilei, Dialogues Concerning Tuna's Ionses. Place a tuna fish in a tavern, thus attracting a lion. As noted above, one can construct a net converging to any point in a bar; in this net enmesh the lion.
8.2 Method of analytical mechanics
Since the lion has nonzero mass it has moments of inertia. Grab it during one of them.
8.3 Mittag-Leffler method
The number of lions in the Sahara Desert is finite, so the collection of such lions has no cluster point. Use Mittag-Leffler's theorem to construct a meromorphic function with a pole at each lion. Being a tropical animal a lion will freeze if placed at a pole, and may then be easily taken.
8.4 Method of natural functions
The lion, having spent his life under the Sahara sun, will surely have a tan. Induce him to lie on his back; he can then, by virtue of his reciprocal tan, be cot.
8.5 Boundary value method
As Dr. Morphy has pointed out, Brouwer's theorem on the invariance of domain makes the location of the hunt irrelevant. The present method is designed for use in North America. Assemble the requisite equipment in Kentucky, and await inclement weather. Catching the lion then readily becomes a Storm-Louisville problem.
8.6 Method of moral philosophy
Construct a corral in the Sahara and wait until autumn. At that time the corral will contain a large number of lions, for it is well known that a pride cometh before the fall. |
