8. MATHEMATICS

1

Number theory, statistical biology and game theory are some of the branches of mathematics that Science Talent Search winner John N. Mather of Princeton, New Jersey, enjoys exploring. His project report, "Nine Postulates for Euclidean Geometry," was part of his winning entry in the Nineteenth Science Talent Search. John's very thorough paper has been abstracted here.

"Introduction

The purpose of this paper is to develop a set of postulates for Euclidean geometry, and to show two things about these postulates: that they actually are true of what we normally call Euclidean space, and that they describe only Euclidean space.

In accordance with these aims, parts of this paper are written in two different ways. When theorems are proved from the postulates, the vocabulary is limited to terms pre­viously defined and a basic English vocabulary to connect these terms in sentences. When it is shown that the postulates actually are true of what is normally called Euclidean space, the vocabulary is not limited in this manner.

In this system of postulates, primitive terms are point and an undefined relation, written ab‹cd for points a,b,c,d. Al­though the relation ab‹cd is not defined, it may be inter­preted as equivalent to the statement "b is less distant from a than d is from c." Such an interpretation is consistent with the given postulates.

Properties of the Undefined Relation

The first three postulates are as follows: (In general, a vertical through the symbol for a relation will mean that the relation does not hold.)

Postulate 1.    If a,b,c are points then ab<cc.     .

only Euclidean geometry. To do this a set of coordinates could be set up to show that the rules of analytic geometry hold, analytic geometry being chosen because it is the most widely known form of geometry and therefore the most acceptable for showing what the postulates in this article describe. Such a process would be lengthy, involving developing theorems about angles and particularly about perpendiculars. It would also necessitate proving that through any point outside of a given line L there is one and only one line parallel to L and in the same plane as it is in. The author is certain, however, that this process can be carried out. .. ."

2

A graphic description of how a project can grow and expand into future work is abstracted here from the paper on "Some Investigations into the Relationship between Symbolic Logic and Electronic Switching Networks," which helped Edward Ganz of Bethesda, Maryland, to become one of the top forty winners of the Eighteenth Science Talent Search.

"Introduction

For many years, I have been interested in the field of mathematics. During the course of my reading, I encountered references to symbolic logic which immediately aroused my curiosity. This relatively new field seemed to offer numerous opportunities for individual research and investigations. Then I came across Claude Shannon's monumental work on the relationship of symbolic logic to electrical switching circuitry. As I had long been interested in electronics, the union of these two fields was particularly attractive to me.

In addition to my explorations into the purely mathematical aspects of symbolic logic, I directed my study to its use in the synthesis and analysis of switching networks and, conversely, the application of switching networks to pure logic. . . .

Progress has been made in showing that symbolic logic is a powerful tool for the analysis and synthesis of electronic switching systems. Little has been done, however, with the converse, namely, the application of electronic systems to the field of symbolic logic. During the course of my research, I became particularly interested in this second relationship. I decided to design and construct an electronic device which would provide extremely rapid solutions to complex problems in symbolic logic. To my knowledge, only two "logic machines" have been built: one at Manchester, England, and the other at M.I.T. I was able to learn little of these machines other than the fact that both were electromechanical. As a result, I proceeded on my own.

My prime interest was in the area of system development of the device as a logical unit. In other words, I was not at this time directly concerned with the design of, for example, a more efficient flip-flop circuit. As a result, by changing the values of the circuit parameters to meet my specific needs, I was able to utilize several tried and proven circuit configura­tions for my new applications. Several subassemblies presented rather unique requirements, however, and I decided to design these myself.

After much thought and experimentation, I arrived at the device described in the following pages. I have called it an electronic logical truth analyzer. Flexible, almost entirely auto­matic and unique in many respects, this device will provide the solution to literally millions of functions of up to 5 logical variables in a maximum of 32 milliseconds.

Logic Circuitry

The logical analyses performed by the machine are carried out by the process of electronic analogy. In effect, that part of the device termed logic circuitry is an electronic switching network whose operation is defined by the logical expressions under consideration. Each fundamental element of this net­work is the electronic representation of the mathematical expression which describes its function.

I decided upon the four logical constants "not," "and," "or" and "if ... then . . ." as the elemental building blocks of the logic circuitry. They are both logically and electronically universal. Logically, these units may be used to build up complex system of thought containing practically all commonly used functions. Electronically, I have taken care of loading to the point where each block will drive any other, and a great number of them may be interconnected without invalid opera­tion. (See Figures 5 and 6.)

System Operation

For the sake of clarity, I will explain the system operation in terms of a simple function of but two variables.

Consider the expression a~b. The operator would like to know for what assigned truth values of "a" and "b" the expres­sion is valid, for what values it is invalid. The problem is introduced to the machine by means of the plugboard. Using patch cords, the operator connects one of the inputs of an "and" element to the variable generator output "a." To the input of a "not" circuit he links one of the "b" outputs of the variable generator. The output of the "not" unit is then con­nected to the other input of the "and" element. A cord is used to join the output of the "and" unit and the input of the amplitude discriminator.

With the throwing of the "operate" switch, the machine begins its automatic analysis. The variable generator sequen­tially assigns all the combinations of truth value to the vari­ables, "a" and "b." Logically operating on these values, the elements representing the constants "not" and "and" produce a pulse pattern which varies as that function of "a" and "b" which is "a<~b."

After passing through the amplitude discriminator, which standardizes the pulse amplitudes, the pulse train is transferred to the cathode-ray display. Synchronized with the variable generator, this output device writes the truth table evaluation defining the expression "a<~b." Because there are only two variables, the solution appears eight times. In other words, the truth values of "c," "d" and "e" have no effect upon the solution.

The simple example explained above serves to illustrate how the machine attacks much more complex problems in­volving more variables and much more intricate relations between them. (See Figure 17.)

The manner in which the machine handles a more complex problem is graphically represented in Figures 18 and 19.

The over-all operation of the machine's units is presented in a general flow diagram. (See Figure 20.)

Photographs of the machine are included to show construc­tion and appearance. (See Figure 21.)

Projections

My reading in symbolic logic, its relationship to electronic switching systems and allied fields, as well as my recent investi­gations in these areas, indicate to me that there are many avenues open for interestingly new research. Some investiga­tions in which I am currently engaged, as well as several of my ideas for future work, follow.

1. am currently engaged in the development of electronic analogs for several of the more obscure logical functions.
2. I have begun work in the design of circuit elements which will enable my logic analyzer to operate on quantified propositions.
   
3. As a result of some rather recent further contemplation, a new system for the electronic solution of logical problems has come to me. I plan to further investigate this system which, through the use of a memory, features sequential, rather than simultaneous, truth evaluation.
   
4. Another area in which I hope to pursue some research is the application of symbolic logic to the analysis of graphical and pictorial data. I hope to relate this to electronic logic systems through the use of flying spot scanning techniques. I am especially interested in working on the problem of image recognition.
5. A rather recent development in symbolic logic is the multivalued logic, where the number of truth values that the variables may assume is greater than two. These systems have some very unusual properties, and it appears that they will be useful for such applications as the analysis of circuits con­
   taining various forms of nonlinear devices. I have done some preliminary work which leads me to believe that it will be possible to deal with these truth-value systems electronically. I hope to do work which will result in formulating the bases of a device for the electronic simulation and analysis of recti­
   fier, and possibly saturable reactor, circuitry."
   

3

"An Investigation of the Packing of Convex Congruent Polygons" was part of the entry of Robert T. Moore of Silver Spring, Maryland, one of the winners of the Fifteenth Science Talent Search. Bob now has completed his undergraduate work in college and is beginning graduate study in theoretical physics. His summers have been spent at the National Bureau of Standards working on mathematical linguistics, artificial intelligence and information retrieval projects in the Data Processing division.

"Introduction

A study of the problem of the packing of convex congruent polygons is roughly analogous to a determination of the possible shapes that the pieces of a jigsaw puzzle may have if they must be alike, straight-sided and convex. The assembled puzzle is unbounded, and the number of polygons in it unlimited.

This problem is a special case of a larger one, the packing of polygons in general. . . .

I have investigated the special case first because it is suffi­ciently limited to make a solution conceivable, although even this is potentially a very broad problem.

A Topological Limitation

Fortunately the topological nature of networks makes the problem much more limited than it appears. On the basis of this nature of networks, as expressed in special cases of Euler's general formula for polyhedra and the Lebesgue-Brouwer "tiling" theorem, I proved the following limiting theorem.

Theorem I: Convex congruent polygons will pack only if they have six or fewer sides. (See appendix I for proof.)

The problem is thus limited to four types of polygons: triangles, quadrilaterals, pentagons and hexagons.

The Oriented Cell Packing

To investigate the problem further it is necessary to formu­late a method of proving whether or not a given polygon will pack. To serve this purpose I evolved and proved:

Theorem II: Any convex polygon which alone or com­bined with congruent polygons forms one large polygon made up of three pairs of opposite sides, such that the members of each pair are parallel and congruent, will pack. (See appendix II for proof.)

This large polygon is often concave and thus at least one of its pairs of sides is made up of broken lines. Parallel, as used in this case, means that all the corresponding parts are parallel; and, combined with the congruency, means that the sides are translatable into one another without changing the orientation of the polygon; hence the name "oriented cell."

After having finished a good portion of this project, I learned that a proof of the oriented cell, somewhat simpler than mine, was evolved in crystallography research, using a non-cartesian coordinate system and vectorial sums in such coordinates. I am using mine, however, because it is my own work and it is proved by the plane geometry techniques with which I am familiar.

Subsequent work is further simplified by two corollaries:

Corollary A: The oriented cell packs as a hexagon made up of the vectorial sums of the parallel and congruent side pairs, such that these vectorial sums are parallel and equal. (See appendix III.)

Corollary B: A polygon with three pairs of sides such that members of each pair are opposite and properly congruent, and one pair parallel, is an oriented cell. (Properly congruent means that one member need not be turned on its back to be proven congruent to the other.)  (See appendix IV.)

It might be noted that although it is possible to prove that all polygons covered by Theorem II and Corollary B will pack, and that those not covered will not pack in an oriented way, (see appendix V), it is not possible to prove that those not covered will not pack at all, without constructing a complete theory of the packing of concave polygons; and this last I leave to later years or wiser heads.

A Method of Deriving Polygons for Study

In order to study polygons and the manner in which they pack, it is desirable to develop a system, rather than to use mere random selection of subjects. The technique I have de­veloped makes use of the fact that, in hexagon packings, exactly three polygons, thus three angles, must lie about any given vertex. Considering one vertex, there are seven sets of angle packings: (1) three successive angles about a point, (2) two successive angles and one alternate angle, (3) three alternate angles, (4) two like angles and a successive, (5) two like angles and one alternate, (6) two like angles and an opposite and (7) three like angles.

My approach has been to consider each of these sets by placing the three prescribed angles about a point, then deter­mining various sets of side equalities necessary for this pack­ing to be possible. For each set I found that there are eight cases of side equalities to be considered. In some cases, I have found that it is necessary to consider four intersection points instead of one, in order to prove that the polygons pack. These four may be considered in a snowflake pattern. (Appendix VI.) In several very stubborn cases, it is necessary to consider still more vertex intersections.

Classes of Packable Polygons

Using the techniques outlined in the last section, it is possible to generate packable polygons (if the technique is carried to enough steps). I have tried to place all of those generated in general classes, and have found and proved three.

The first packing grows out of Set 1 and is quite general. The hexagon considered has one pair of opposite sides parallel and equal, and two of these hexagons may be combined to form an oriented cell. All Set 1 hexagons are covered in this class. (See appendix Vila.) It might be noted that if one unlimited side of this hexagon is of no length, a class of pen­tagons is generated. If the parallel and equal sides are both of no length, the class covers all quadrilaterals. If one of the sides of the quadrilateral is of no length, this class also covers all triangles. It is also the most general class, having a seven­fold infinity of polygons satisfying the conditions. (Appendix VII B.)

The second class of packings grows out of Set 2, and has one pair of opposite sides equal, the two sides adjacent to one member of the first pair also equal, and three particular angles adding up to 360°. Four of these make up an oriented cell. (See appendix VIII A.) It is a more limited class than Class I, having only a six-fold infinity of satisfying polygons, (appen­dix VIII B), and covers most of the Set 2 packings which do not fit in Class I.

The third class grows from Set 3, and has three separate pairs of adjacent sides equal, the angles included by these pairs of sides all equal to 120°. Three of these polygons make up an oriented cell. It covers two of the Set 3 cases, the others being Class I polygons. (See appendix IX A.) It is a very limited class, having but a three-fold infinity of polygons. (Appendix DCB.)

This completes the classes I have thus far proved.

Work Yet To Be Done

There is considerable scope for further study of this problem.

I hope to complete the remaining studies as soon as possible.

Possible Applications

If applications of this work are considered desirable, they may be found. In its three dimensional counterpart, this study is possibly relevant to crystallography and organic chemistry (high polymers) and may have applications to problems yet unposed. . . ."

points x and y." "||=" means "both parallel and con­gruent."

Given: Polygon A...B...C...D...E...F.., such that A ... B||D ... E, B... ClIssE.. .F, C...D||=sF...A.

Prove: Hexagon ABCDEF has AB||=DE, BC||=EF, CD||=FA.

Proof:

1. A... BsD ... E, .'. they may be superimposed.
   
2. By identity, AB=DE.
   
3. In the cell each component of A... B is || and = to a corresponding component of D . . . E, AB is the vectorial sum of A ... B, and DE is the vectorial sum of D... E, .'.AB||DE.
4. \ABllssDE.
5. By similar method, BC||=FE, CD||=AF.
   

Appendix VI

In fitting three angles about a point (as in figure 3) a number of sets of side equalities may be derived. Each angle is included by two sides, thus there are three side pairs. Each of these side pairs may have two orientations (by turning the angle upside down in its space), thus the number of possible sets is 23 or 8.

At times the equalities thus derived, coupled with the in­formation about side orientation gained from the angles (which must add up to 360°), are not sufficient to prove that a given polygon packs as one of the general classes. In this case, four or more vertices must be considered, normally in an arrangement such as shown in figure 4."

Are You Ready To Move Onto The Next Lesson? Click Here….