Archive for the ‘Geometry’ Category

Simple ideas, complex possibilities, elegant and beautiful results

February 11, 2011

Possibilities of great subtlety, elegance, and power can follow from the simplest ideas. Leonardo da Vinci is often credited with offering a variation on this theme, but the basic idea is much older. Philosophy, for instance, began with Plato’s distinction between name and concept. This realization that words are not the things they stand for has informed and structured each of several scientific revolutions.

How so? It all begins from the reasons why Plato required his students to have studied geometry. He knew that those familiar with the Pythagorean theorem would understand the difference between any given triangle and the mathematical relationships it represents. No right triangle ever definitively embodies a perfect realization of the assertion that the square of the hypotenuse equals the sum of the squares of the other two sides. The mathematical definition or concept of a triangle is not the same thing as any actual triangle.

The subtlety and power of this distinction became apparent in its repeated application throughout the history of science. In a sense, astronomy is a geometry of the heavens, Newton’s laws are a geometry of gravity, Ohm’s law is a geometry of electromagnetism, and relativity is a geometry of the invariance of mass and energy in relation to the speed of light. Rasch models present a means to geometries of literacy, numeracy, health, trust, and environmental quality.

We are still witnessing the truth, however partial, of Whitehead’s assertion that the entire history of Western culture is a footnote to Plato. As Husserl put it, we’re still struggling with the possibility of creating a geometry of experience, a phenomenology that is not a mere description of data but that achieves a science of living meaning. The work presented in other posts here attests to a basis for optimism that this quest will be fruitful.

Geometrical and algebraic expressions of scientific laws

April 12, 2010

Geometry provides a model of scientific understanding that has repeatedly proven itself over the course of history. Einstein (1922) considered geometry to be “the most ancient branch of physics” (p. 14). He accorded “special importance” to his view that “all linear measurement in physics is practical geometry,” “because without it I should have been unable to formulate the theory of relativity” (p. 14).

Burtt (1954) concurs, pointing out that the essential question for Copernicus was not “Does the earth move?” but, rather, “…what motions should we attribute to the earth in order to obtain the simplest and most harmonious geometry of the heavens that will accord with the facts?” (p. 39). Maxwell similarly employed a geometrical analogy in working out his electromagnetic theory, saying

“By referring everything to the purely geometrical idea of the motion of an imaginary fluid, I hope to attain generality and precision, and to avoid the dangers arising from a premature theory professing to explain the cause of the phenomena. If the results of mere speculation which I have collected are found to be of any use to experimental philosophers, in arranging and interpreting their results, they will have served their purpose, and a mature theory, in which physical facts will be physically explained, will be formed by those who by interrogating Nature herself can obtain the only true solution of the questions which the mathematical theory suggests.” (Maxwell, 1965/1890, p. 159).

Maxwell was known for thinking visually, once as a student offering a concise geometrical solution to a problem that resisted a lecturer’s lengthy algebraic efforts (Forfar, 2002, p. 8). His approach seemed to be one of playing with images with the aim of arriving at simple mathematical representations, instead of thinking linearly through a train of analysis. A similar method is said to have been used by Einstein (Holton, 1988, pp. 385-388).

Gadamer (1980) speaks of the mathematical transparency of geometric figures to convey Plato’s reasons for requiring mathematical training of the students in his Academy, saying:

“Geometry requires figures which we draw, but its object is the circle itself…. Even he who has not yet seen all the metaphysical implications of the concept of pure thinking but only grasps something of mathematics—and as we know, Plato assumed that such was the case with his listeners—even he knows that in a manner of speaking one looks right through the drawn circle and keeps the pure thought of the circle in mind.” (p. 101)

But exactly how do geometrical visualizations lend themselves to algebraic formulae? More specifically, is it possible to see the algebraic structure of scientific laws in geometry?

Yes, it is. Here’s how. Starting from the Pythagorean theorem, we know that the square of a right triangle’s hypotenuse is equal to the sum of the squares of the other two sides. For convenience, imagine that the lengths of the sides of the triangle, as shown in Figure 1, are 3, 4, and 5, for sides a, b, and c, respectively. We can count the unit squares within each side’s square and see that the 25 in the square of the hypotenuse equal the sum of the 9 in the square of side a and the 16 in the sum of side b.

That mathematical relationship can, of course, be written as

a2 + b2 = c2

which, for Figure 1, is

32 + 42 = 52 = 9 + 16 = 25

Now, most scientific laws are written in a multiplicative form, like this:

m = f / a

or

f = m * a

which, of course, is how Maxwell presented Newton’s Second Law. So how would the Pythagorean Theorem be written like a physical law?

Since the advent of small, cheap electronic calculators, slide rules have fallen out of fashion. But these eminently useful tools are built to take advantage of the way the natural logarithm and the number e (2.71828…) make division interchangeable with subtraction, and multiplication interchangeable with addition.

That means the Pythagorean Theorem could be written like Newton’s Second Law of Motion, or the Combined Gas Law. Here’s how it works. The Pythagorean Theorem is normally written as

a2 + b2 = c2

but does it make sense to write it as follows?

a2 * b2 = c2

Using the convenient values for a, b, and c from above

32 + 42 = 52

and

9 + 16 = 25

so, plainly, simply changing the plus sign to a multiplication sign will not work, since 9 * 16 is 144. This is where the number e comes in. What happens if e is taken as a base raised to the power of each of the parameters in the equation? Does this equation work?

e9 * e16 = e25

which, substituting a for e9, b for e16, and c for e25, could be represented by

a * b = c

and which could be solved as

8103 * 8,886,015 ≈ 72,003,378,611

Yes, it works, and so it is possible to divide through by e16 and arrive at the form of the law used by Maxwell and Rasch:

8103 ≈ 72,003,378,611 / 8,886,015

or

e9 = e25 / e16

or, again substituting a for e9, b for e16, and c for e25, could be represented by

a = c / b

which, when converted back to the additive form, looks like this:

a = c – b

and this

9 = 25 – 16 .

Rasch wrote his model in the multiplicative form of

εvi = θvσi

and it is often written in the form of

Pr {Xni = 1} = eβnδi / 1 + eβnδi

or

Pni = exp(Bn – Di) / [1 + exp(Bn – Di)]

which is to say that the probability of a correct response from person n on item i is equal to e taken to the power of the difference between the estimate β (or B) of person n‘s ability and the estimate δ (or D) of item i‘s difficulty, divided by one plus e to that same power.

Logit estimates of Rasch model parameters taken straight from software output usually range between ­-3.0 or so and 3.0. So what happens if a couple of arbitrary values are plugged into these equations? If someone has a measure of 2 logits, what is their probability of a correct answer on an item that calibrates at 0.5 logits? The answer should be

e2-0.5 / (1 + e2-0.5).

Now,

e1.5 = 2.718281.5 = 4.481685….

and

4.481685 / (1 + 4.481685) ≈ 0.8176

For a table of the relationships between logit differences, odds, and probabilities, see Table 1.4.1 in Wright & Stone (1979, p. 16), or Table 1 in Wright (1977).

This form of the model

Pni = exp(Bn – Di) / [1 + exp(Bn – Di)]

can be rewritten in an equivalent form as

[Pni / (1 – Pni)] = exp(Bn – Di) .

Taking the natural logarithm of the response probabilities expresses the model in perhaps its most intuitive form, often written as

ln[Pni / (1 – Pni)] = Bn – Di .

Substituting a for ln[Pni / (1 – Pni)], b for Bn, and c for Di, we have the same equation as we had for the Pythagorean Theorem, above

a = c – b .

Plugging in the same values of 2.0 and 0.5 logits for Bn and Di,

ln[Pni / (1 – Pni)] = 2.0 – 0.5 = 1.5.

The logit value of 1.5 is obtained from response odds [Pni / (1 – Pni)] of about 4.5, making, again, Pni equal to about 0.82.

Rasch wrote the model in working from Maxwell like this:

Avj = Fj / Mv .

So when catapult j’s force F of 50 Newtons (361.65 poundals) is applied to object v’s mass M of 10 kilograms (22.046 pounds), the acceleration of this interaction is 5 meters (16.404 feet) per second, per second. Increases in force relative to the same mass result in proportionate increases in acceleration, etc.

The same consistent and invariant structural relationship is posited and often found in Rasch model applications, such that reasonable matches are found between the expected and observed response probabilities are found for various differences between ability, attitude, or performance measures Bn and the difficulty calibrations Di of the items on the scale, between different measures relative to any given item, and between different calibrations relative to any given person. Of course, any number of parameters may be added, as long as they are included in an initial calibration design in which they are linked together in a common frame of reference.

Model fit statistics, principal components analysis of the standardized residuals, statistical studies of differential item/person functioning, and graphical methods are all applied to the study of departures from the modeled expectations.

I’ve shown here how the additive expression of the Pythagorean theorem, the multiplicative expression of natural laws, and the additive and multiplicative forms of Rasch models all participate in the same simultaneous, conjoint relation of two parameters mediated by a third. For those who think geometrically, perhaps the connections drawn here will be helpful in visualizing the design of experiments testing hypotheses of converging yet separable parameters. For those who think algebraically, perhaps the structure of lawful regularity in question and answer processes will be helpful in focusing attention on how to proceed step by step from one definite idea to another, in the manner so well demonstrated by Maxwell (Forfar, 2002, p. 8). Either way, the geometrical and/or algebraic figures and symbols ought to work together to provide a transparent view on the abstract mathematical relationships that stand independent from whatever local particulars are used as the medium of their representation.

Just as Einstein held that it would have been impossible to formulate the theory of relativity without the concepts, relationships, and images of practical geometry, so, too, may it one day turn out that key advances in the social and human sciences depend on the invariance of measures related to one another in the simple and lawful regularities of geometry.

Figure 1. A geometrical proof of the Pythagorean Theorem

References

Burtt, E. A. (1954). The metaphysical foundations of modern physical science (Rev. ed.) [First edition published in 1924]. Garden City, New York: Doubleday Anchor.

Einstein, A. (1922). Geometry and experience (G. B. Jeffery, W. Perrett, Trans.). In Sidelights on relativity (pp. 12-23). London, England: Methuen & Co. LTD.

Forfar, J. (2002, June). James Clerk Maxwell: His qualities of mind and personality as judged by his contemporaries. Mathematics Today, 38(3), 83.

Gadamer, H.-G. (1980). Dialogue and dialectic: Eight hermeneutical studies on Plato (P. C. Smith, Trans.). New Haven: Yale University Press.

Holton, G. (1988). Thematic origins of scientific thought (Revised ed.). Cambridge, Massachusetts: Harvard University Press.

Maxwell, J. C. (1965/1890). The scientific papers of James Clerk Maxwell (W. D. Niven, Ed.). New York: Dover Publications.

Wright, B. D. (1977). Solving measurement problems with the Rasch model. Journal of Educational Measurement, 14(2), 97-116 [http://www.rasch.org/memo42.htm].

Wright, B. D., & Stone, M. H. (1979). Best test design: Rasch measurement. Chicago, Illinois: MESA Press.

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