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Curves for space and time, continued

The following is a continuation and revision of the previous post, here.

Based on the differential geometry part of the book Shape Interrogation for Computer Aided Design and Manufacturing by Nicholas M. Patrikalakis and Takashi Maekawa of MIT. A pdf version in parallel is here.

Let a three-dimensional curve be expressed in parametric form as x = x(t); y = y(t); z = z(t); where the coordinates of the point (x, y, z) of the curve are expressed as functions of a parameter t (time) within a closed interval t1tt2. The functions x(t), y(t), and z(t) are assumed to be continuous with a sufficient number of continuous derivatives.

In vector notation the parametric curve can be specified by a vector-valued function r = r(t), where r represents the position vector (i.e., r(t) = (x(t), y(t), z(t)).

curve1

Displacement Δr connecting points A and B on parametric curve r(t).

Consider a segment (displacement) of a parametric curve r = r(t) between two points P(r(t)) and Q(r(tt)) as shown in the figure above. As point B approaches A or in other words Δt → 0, the length s becomes the differential arc length of the curve:

ds = |dr/dt| dt = | rt | dt = √(rtrt) dt,

where the superscript t denotes differentiation with respect to the arc time parameter t. The vector rt = dr/dt is called the tangent vector at point A.

Then the arc length, s, of a segment of the curve between points r(t0) and r(t) can be obtained as follows:

s(t) = ∫ ds = ∫ √(rtrt) dt = ∫ √((dx/dt)2 + (dy/dt)2 + (dz/dt)2) dt.

The magnitude of the tangent vector is

| rt | = ds/dt = v.

Hence the unit tangent vector is

Ts = rt / | rt | = (dr/dt) / (ds/dt) = dr/dsrs,

where the superscript s denotes differentiation with respect to the arc length parameter, s.

If r(s) is an arc length parametrized curve, then rs(s) is a unit vector, and hence rsrs = 1. Differentiating this relation, we obtain rsrss = 0, which states that rss is orthogonal to the tangent vector, provided it is not a null vector. The unit vector

Ns = rss(s) / |rss(s)| = Tss(s)/|Tss(s)|,

which has the direction and sense of rss(s) is called the unit principal normal vector at s. The plane determined by the unit tangent and normal vectors Ts(s) and Ns(s) is called the osculating plane at s. The curvature is

κs ≡ 1/ρ = |rss(s)|,

and its reciprocal ρ is called the radius of curvature at s. It follows that

rss = Tss = κs Ns.

The vector ks = rss = Tss is called the curvature vector, and measures the rate of change of the tangent along the curve. By definition κs is nonnegative, thus the sense of the normal vector is the same as that of rss(s). For a three-dimensional curve, the curvature is

κs = |rt × rtt| / | rt |³.


Let a three-dimensional curve be expressed in parametric form as X = X(s); Y = Y(s); Z = Z(s); where the coordinates of the point (X, Y, Z) of the curve are expressed as functions of a parameter s (length) within a closed interval s1ss2. The functions X(s), Y(s), and Z(s) are assumed to be continuous with a sufficient number of continuous derivatives.

In vector notation the parametric curve can be specified by a vector-valued function w = w(s), where w represents the position vector (i.e., w(s) = (X(s), Y(s), Z(s)).

curve2

Distimement Δw connecting points C and D on parametric curve w(s).

Consider a segment (distimement) of a parametric curve w = w(s) between two points C(w(s)) and D(w(ss)) as shown in the figure above. As point D approaches C or in other words Δs → 0, the length t becomes the differential arc time of the curve:

dt = |dw/ds| ds = | ws | ds = √(wsws) ds,

where ws = dw/ds, which is called the tangent vector at point C. Then the arc time, t, of a segment of the curve between points w(s0) and w(s) can be obtained as follows:

t(s) = ∫ dt = ∫ √(wsws) ds = ∫ √((dX/ds)2 + (dY/ds)2 + (dZ/ds)2) ds.

The vector ws = dw/ds is called the tangent vector at point C. The magnitude of the tangent vector is

| ws | = dt/ds = u.

Hence the unit tangent vector is

Ttws / | ws | = (dw/ds) / (dt/ds) = dw/dtwt.

If w(t) is an arc length parametrized curve, then wt(t) is a unit vector, and hence wtwt = 1. Differentiating this relation, we obtain wtwtt = TtTtt = 0, which states that wtt is orthogonal to the tangent vector, provided it is not a null vector. The unit vector

Nt = wtt(t) / |wtt(t)| = Ttt(t)/|Ttt(t)|,

which has the direction and sense of wtt(t) is called the unit principal normal vector at t. The plane determined by the unit tangent and normal vectors Tt(t) and Nt(t) is called the osculating plane at t. The curvature is

κt ≡ 1/ρ = |wtt(t)| = |Ttt(t)|,

and its reciprocal ρ is called the radius of curvature at t. It follows that

wttTtt = κt Nt.

The vector kt = wttTtt is called the curvature vector, and measures the rate of change of the tangent along the curve. By definition κ is nonnegative, thus the sense of the normal vector is the same as that of wtt(t). For a three-dimensional curve, the curvature is

κt = |ws × wss| / | ws |³.


Here are some useful formulae of the derivatives of arc length, s, and the arc time, t:

v = st = ds/dt = | rt | = (rtrt)1/2 = 1/| ws | = 1/(wsws)1/2,

a = stt = dst/dt = (rtrtt) / (rtrt)1/2 = – (wswss) / (wsws)4/2,

sttt = dstt/dt = [(rtrt)(rtrttt + rttrtt) – (rtrtt)²] / (rtrt)3/2

= – [(wsws)(wswsss + wsswss) – 4(wswss)²] / (wsws)7/2,

u = ts = dt/ds = 1/| rt | = 1/(rtrt)1/2 = | ws | = (wsws)1/2,

b = tss = dts/ds = – (rtrtt) / (rtrt)4/2 = (wswss) / (wsws)1/2,

tsss = dtss/ds = – [(rtrt)(rtrttt + rttrtt) – 4(rtrtt)²] / (rtrt)7/2

= [(wsws)(wswsss + wsswss) – (wswss)²] / (wsws)3/2.

Curves for space and time

The following is slightly modified from the differential geometry part of the book Shape Interrogation for Computer Aided Design and Manufacturing by Nicholas M. Patrikalakis and Takashi Maekawa of MIT.

A plane curve can be expressed in parametric form as x = x(t); y = y(t); where the coordinates of the point (x, y) of the curve are expressed as functions of a parameter t (time) within a closed interval t1tt2. The functions x(t) and y(t) are assumed to be continuous with a sufficient number of continuous derivatives. The parametric representation of space curves is: x = x(t); y = y(t); z = z(t); t1tt2.

In vector notation the parametric curve can be specified by a vector-valued function r = r(t), where r represents the position vector (i.e., r(t) = (x(t), y(t), z(t)).

curve

Displacement Δr connecting points P and Q on parametric curve r(t).

Let us consider a segment (displacement) of a parametric curve r = r(t) between two points P(r(t)) and Q(r(tt)) as shown in the figure above. As point Q approaches P or in other words Δt → 0, the length s becomes the differential arc length of the curve:

ds = |dr/dt| dt = | r | dt = √(rr) dt.

Here the dot denotes differentiation with respect to the parameter t. Therefore the arc length of a segment of the curve between points r(t0) and r(t) can be obtained as follows:

s(t) = ∫ ds = ∫ √(rr) dt = ∫ √(x2(t) + y2(t) + z2(t)) dt.

The vector dr/dt is called the tangent vector at point P. The magnitude of the tangent vector is

| r | = ds/dt = v.

Hence the unit tangent vector becomes

T = r / | r | = (dr/dt) / (ds/dt) = dr/ds.

Here the prime ¹ denotes differentiation with respect to the arc length, s. We list some useful formulae of the derivatives of arc length s with respect to parameter t and vice versa:

v = s = ds/dt = | r | = (rr)1/2,

a = s•• = ds/dt = (rr) / (rr)1/2,

s = ds/dt = [(rr)(rr + rr) – (rr)²] / (rr)3/2,

u = t¹ = dt/ds = 1/| r | = 1/(r • r)1/2,

b = t¹¹ = d/ds = – (rr) / (rr)4/2,

t¹¹¹ = dt¹¹/ds = – [(rr)(rr + rr) – 4(rr)²] / (rr)7/2.

If r(s) is an arc length parametrized curve, then (s) is a unit vector, and hence = 1. Differentiating this relation, we obtain r¹¹ = 0, which states that r¹¹ is orthogonal to the tangent vector, provided it is not a null vector.

The unit vector

N = r¹¹(s)/|r¹¹(s)| = (s)/|(s)|,

which has the direction and sense of (s) is called the unit principal normal vector at s. The plane determined by the unit tangent and normal vectors T(s) and N(s) is called the osculating plane at s. The curvature is

κ ≡ 1/ρ = |r¹¹(s)|,

and its reciprocal ρ is called the radius of curvature at s. It follows that

r¹¹ = = κN.

The vector k = r¹¹ = is called the curvature vector, and measures the rate of change of the tangent along the curve. By definition κ is nonnegative, thus the sense of the normal vector is the same as that of r¹¹(s).

For a space curve, the curvature is

κ = |r × r| / |r|³.

Interpretation of math and science

There’s a common understanding that most writings need to be interpreted — especially those of a religious or philosophical nature. But mathematical and scientific writings are similar and need to be interpreted, too.

Consider that mathematicians and scientists write as if they were creating a world. Mathematicians say things like, “Let there be a line and a point not lying on it such that …” Or scientists will say, “Occam’s razor is a principle of science” as if they can assert principles ex nihilo. How should these creations be interpreted?

Mathematicians write as if infinity were next door: “As x approaches infinity …” Scientists write as if the entire universe were in view: “The universal theory of gravitation states …” But universal theories turn out to have limitations. And the One who is actually infinite never appears in mathematics. So what do these locutions really mean?

Before the discovery or invention (which one?) of non-Euclidean geometry and its application to physics, it was common for people to think that Euclidean geometry described the space we live in. It is said that most mathematicians are Platonists, and believe that mathematical entities literally exist. Since the 19th century, the literal interpretation of science has been in ascendancy, in which nature is all that exists (i.e., scientific naturalism, see here).

Some say modern science was an unintended consequence of the Reformation’s rejection of levels of meaning in the Bible, which led to a more literal interpretation of God’s other book, the book of nature. The conclusion from all this is that mathematics and science need to be interpreted as much as religious or philosophical writings. What’s your interpretation?

The story of nothing

Mathematics is the study of nothing. We make something out of nothing, acting the creator in a world of nothing. Here’s the story:

In the beginning is nothing. Not totally nothing because we’re there. But a blank page, a clear slate, a tabula rasa.

We draw a distinction, a part of nothing. The indistinct blankness of nothing gains a something. We indicate the something. We indicate the original nothing. We develop a logic of nothing.

We draw a place of nothing, a point. We draw a line of points, then a plane, and a solid. We select an original nothing, an origin. We develop a geometry of nothing.

We draw a number of nothing, a zero. We add it, subtract it, and multiply it. We raise numbers to its power. We develop an arithmetic and algebra of nothing.

We reduce a number to nothing, an infinitesimal. We define a function of it. We take its tangent, its sum, and its mean. We develop a calculus of nothing.

We take the set of nothing, the null set. We intersect it, union it, complement it. We take the power set of it. We find its cardinality. We develop a set theory of nothing.

In the end we have nothing, nothing but mathematics.

Laws of form

The remarkable book Laws of Form by George Spencer-Brown was published in 1969 and is almost forgotten today. The best expositors have been William Bricken with his boundary mathematics, Louis Kauffman with his knot theory, and Francisco Varela with his work on self-reference. Otherwise it has become something of an underground classic but otherwise forgotten. There are several reasons for the latter, including the exaggerated claims of the author and some enthusiasts. That said, I think it’s worth rehabilitating the Laws of Form (LoF) and rightly discerning its significance.

LoF is a work on diagrammatic reasoning in the tradition of Leibniz and CS Peirce. It is a calculus, complete with arithmetic and algebra, based on the act of making and indicating a distinction. Thus it is a work of mathematical realism, which begins to explain why it is not of interest to anti-realists. Its greatest accomplishment is the unified treatment of injunction and indication, of implication and negation via a single symbol, called a cross.

Here are the arithmetic axioms of the calculus of indications:

lof2

That’s it. The inverted “L” is the cross symbol. A cross next to another cross is equal to one cross; this is the Law of Calling. A cross inside another cross is equal to blank, that is, as if no cross had been written. This is the Law of Crossing, hence the name of the symbol, Cross.

This is a two-dimensional calculus, which gives it advantages that one-dimensional notation does not have. It also makes it hard to display typographically. The best alternative is simply to use parentheses or brackets:

( ) ( ) = ( ) and (( )) =  .

These arithmetic axioms can be used to derive two algebraic axioms:

((A) (B)) C = ((A C) (B C)) and ((A) A) =   .

From this a complete calculus can be constructed. It is isomorphic to Boolean algebra and other functionally-complete binary calculi, which is another reason LoF hasn’t stirred a lot of interest.

Things get more interesting as we review where this calculus comes from. Again this exhibits its realism; the standard approach for mathematics and symbolic logic is to begin with algebraic axioms or postulates without reference to any model or reality.

Let’s begin with a blank surface, say a blank page of paper. Now draw a distinction on this surface; that is, draw a closed curve or divide the page into two parts. Notice what has happened: part of the paper is distinguished from the rest of the paper by being to one side of the curve, say the inside. The curve separates the other side from the inside; call it the outside. But the original piece of paper is still there. We can still consider the whole piece of paper.

This process is symbolized by LoF as follows: what is outside the cross (or parentheses) can be seen inside the cross (or parentheses) if we change perspectives to the whole page. This is symbolized in a theorem:

(A) B = (A B) B.

So the distinction that is drawn is not between two contraries but within one space, represented by the whole page. It also shows the distinction can be undermined. This has been exploited to represent self-reference.

Much more could be said about LoF but that’s it for now.

Actual infinity

Before the 19th century it was commonly understood that only God (or perhaps the “gods”) were actually infinite.  If one spoke about the actual infinite, one was doing theology.  In mathematics infinity was considered a manner of speaking, which was clarified in the early 19th century with the careful definition of limits.

In the late 19th century Cantor’s infinite sets were seen as a challenge to this because they treated infinite sets as complete entities.  But there is still no need to consider this essentially different from the relative manner infinity is treated elsewhere in mathematics.

The idea that the universe may be eternal is ancient but there has never been a comprehensive treatment of what this would mean.  Theologians are still struggling to understand in what sense time could exist before the creation.  Nonmetric time seems to be the best solution.

The burden is on anyone who speaks of a physical infinite to explain in detail what they mean.  Otherwise, they’re just throwing words around.

August 2014

Means and Extremes

Means and extremes in classical mathematics have to do with proportions.

If A is to B as C is to D, we write A : B :: C : D.  This is ordered so that A is greater than or equal to B and C is greater than or equal to D.  A and D are called the extremes; B and C are called the means.

By elementary arithmetic the product of the extremes equals the product of the means:

A x D = B x C.

If B = C, then B is the mean proportional or geometric mean of A and D.  In that case B is the positive square root of A x D.

This provides a basic principle for centrism: the means are between the extremes in a principles manner.

The dialectic of extremes and means

The dialectic of extremes and means is a method of reasoning whereby one begins with extremes and reasons to means or vice versa.  If one begins with means, these are considered as unanalyzed entities, attributes, propositions, etc.  The goal is to work out the implications of them as principles or to analyze them into their constituent parts as a combination of extremes.  If one begins with extremes, these are considered as unsynthesized entities, attributes, propositions, etc.  The goal is to synthesize them into their fullness and completion as integrated means or to work from partial truths toward full truths.

We live among means, that is, we live in the middle ground, a mesosphere where things are muddled and messy but familiar and common.  Philosophy is often said to begin here, with what is commonly known rather than with specialized knowledge.  Whatever we find must come back to the middle ground where we live or else it is like a dream unrelated to our lives.

Classical deductive logic works from truths to their implications while preserving truth.  It assumes that truth is known at the beginning, that truths are known in the middle ground of life.  They may be known because they are axiomatic (worthy of assent) or because they are self-evident, or because they were given by a trustworthy source.  The outworking of such truths leads toward extremes.

The dialectic of reasoning from extremes to means is focused on the end, not the beginning.  It does not follow from truths; it leads toward truths.  One does not usually begin with truth.  One usually begins with something at hand, something muddled and messy.  Truth is something that must be sought.  This dialectic begins with partial truths and reasons toward full truth.

Extremes express simple but partial truths.  Proverbial statements often express extremes – that’s why there are often contrary proverbs.  For example, the Book of Proverbs includes these two:  Do not answer a fool according to his folly, or you will be like him yourself.  Answer a fool according to his folly, or he will be wise in his own eyes.  (Pr. 26.4-5)

There are many pairs of entities, attributes, propositions, etc., which express contrary extremes and are partially true.  For example, a preference for simplicity leads to extremes:  in classification and typology, the extremes are all elements in one class and every element in its own class.  Some people (called lumpers) tend to combine elements into fewer classes and others (called splitters) tend to split elements into more classes.  Who is right?  They are both partially right.

Reasoning from extremes to means may be deductive or inductive.  The deductive form works via a form of backward chaining.  It starts with a mean which is a hypothesis or goal and works backwards from the consequent to the antecedent to see if the extremes will support this or any of these consequents.  Instead of reasoning from truth, it is reasoning from partial truths.  The result is a combination of partial truths, which together form a complete truth.

As an illustration of reasoning from extremes to means, consider arithmetic.  Start by defining numbers recursively: if x is a number, then f(x) is a number.  For example, if x is a number then x+1 is a number.  (Addition could be left undefined at this point but let’s assume it is ordinary addition.)  Next, consider the extremes: what are the first last numbers, if they exist?

Answer A:  There is a first number; call it 0.  There is a last number; call it 2, where 2 is 0+1+1.  This is arithmetic modulo 3.

Answer B:  There is a first number.  Call it 0 (or 1, if you prefer).  There is no last number in the sense that there is no unique last number (the sequence must not converge and one can stop at any number arbitrarily).  We conclude that 0+1 is a number, as are 0+1+1, and so on in sequence without end.

Answer C: There is no first number in the sense that there is no unique first number.  There is a last number which depends on the recursion and the arbitrary first number (called the seed number).  The sequence must be convergent.  For example, let the seed number be 1 and the recursion such that if x is a number, then the reciprocal of x+1 is also a number.  This leads to the sequence 1, ½, 2/3, 3/5, 5/8, and so on.  The last number of this sequence is (-1+√5)/2, sometimes called φ (or 1/φ).  Notice that other seed numbers could lead to the same last number.

In these examples the numbers formed by the recursions are the means.  The extremes (those directly stipulated as numbers or used as seed numbers or formed by sequences) are not really numbers.  From ancient times a number has been defined as a multitude so the first number is the second member of the number sequence and there is no last number.  The extreme numbers are the limits of ordinary numbers.  Ordinary numbers are analogous to partial truths, and extreme numbers are analogous to full truths.

These examples lead to the observation that sometimes the extremes may be contrary in different and multiple ways.  First and last are natural extremes but other attributes may be contrary, too:  definite and indefinite, arbitrary and determinate, convergent and divergent, etc.

Conjecture: convergent and divergent sequences may be put into one-to-one correspondence.

November 2013

Alternate arithmetic

A model is a realization of a mathematical formalism.  So ordinary arithmetic is a model of ordinary algebra.  That is, the algebra of the integers, the rational numbers, and the real numbers is realized by the arithmetic of the integers, the rational numbers, and the real numbers, respectively.  Are there other models of ordinary algebra?  Yes.  One alternate model in particular is a simple opposite of ordinary arithmetic and deserves the name alternate arithmetic.

The one-to-one correspondence between ordinary arithmetic and alternate arithmetic is as follows:

Property Ordinary Arithmetic Alternate Arithmetic
Origin 0
Ultimate 0
Unity 1 1
Duality 2 1/2
Left Order < >
Right Order > <
Minimum Digit 0 9
Maximum Digit 9 0
Minimum Decimal …000.000… …999.999…
Maximum Decimal …999.999… …000.000…

What is alternate arithmetic good for?  Ordinary arithmetic implicitly assumes beginning with nothing and adding something.  So the number N means 0+N.  Alternate arithmetic assumes beginning with everything and subtracting something.  So the alternate number N means 1/N.  That is, ordinary arithmetic is additive and alternate arithmetic is subtractive.  The square of opposition in quantification logic presents something similar.  None and some form an additive logic.  All and not all form a subtractive logic.

Instead of using alternate symbols, we may reinterpret the symbols of ordinary arithmetic.  In this way, alternate arithmetic looks exactly like ordinary arithmetic but means something opposite.  Or we may simply write the alternate number 1/N as N by abuse of notation.

November 2013