On means, polynomials and special functions
Johan Gielis
University of Antwerp
Antwerp, Belgium
johan.gielis@ua.ac.be
Rik Verhulst
Roquebrune sur Argens, France
rikverhulst@yahoo.fr
Diego Caratelli
The Antenna Company Nederland B.V.
Rhenen, the Netherlands
diego.caratelli@antennacompany.com
Paolo E. Ricci
Campus Bio-Medico University
Rome, Italy
paoloemilioricci@gmail.com
Ilia Tavkhelidze
Tbilisi State University Tbilisi, Georgia
ilia.tavkhelidze@tsu.ge
Summary
We discuss how derivatives can be considered as a game of cubes and beams,
and of geometric means. The same principles underlie wide classes of polynomials. This results in an unconventional view on the history of the differentiation and differentials.
On derivation and derivatives
In On Proof and Progress in Mathematics William Thurston describes how
people develop an “understanding” of mathematics [1]. He uses the exam1
ple of derivatives, that can be approached from very different viewpoints,
allowing individuals to develop their own understanding of derivatives.
“The derivative can be thought of as:
(1) Infinitesimal: the ratio of the infinitesimal change in the value
of a function to the infinitesimal change in a function.
(2) Symbolic: the derivative of xn is nxn−1 , the derivative of
sin (x) is cos (x), the derivative of f ◦ g is f ◦ g × g , etc...
(3) Logical: f (x) = d if and only if for every there is a δ, such
that 0 < |Δx| < δ when
f(x + Δx) − f(x)
(1)
− d < δ
Δx
(4) Geometric: the derivative is the slope of a line tangent to the
graph of the function, if the graph has a tangent.
(5) Rate: the instantaneous speed of f (t) = d, when t is time.
(6) Approximation: The derivative of a function is the best linear
approximation to the function near a point.
(7) Microscopic: The derivative of a function is the limit of what
you get by looking at it under a microscope of higher and higher
power.
........
(37) The derivative of a real-valued function f in a domain D is
the Lagrangian section of the cotangent bundle T ∗ (D) that gives
the connection form for the unique flat connection on the trivial
R−bundle D × R for which the graph of f is parallel.”
Thurston provides this partial list but states that this list can be still
extended. With respect to ‘individual’ understanding, “one person’s clear
mental image is another person’s intimidation. Human understanding does
not follow a single path, as a computer with a central processing unit; our
brains are much more complex and capable of far more than a single path” [1].
2
In addition, we should not forget that it has taken mathematicians thousands
of years to come to a good understanding of the concept.
In this article we will use (2) and show how this reduces to a game of
cubes and unit elements. The same procedure underlies important special
polynomials in mathematics, as recent research shows. Essentially the game
component of cubes and beams, very clear to mathematicians from the 16th
and 17th century, and the unified approach for polynomials are the same.
Many of the other different definitions of derivatives simply follow from these
observations.
The geometry of means
Throughout the article we will only use very simple arguments, starting from
geometric means, beams and cubes and unit (or neutral) element. To develop
the argument we use a method proposed by the second author, a geometrical representation of the nth −arithmetic, nth −geometric and nth −harmonic
means [2], [3], [4].
For n = 2 the geometric mean GM and arithmetic mean AM are geometry in the sense that they are solutions to one of the oldest optimization
problems: For a given rectangle with sides a and b, AM between a and b
is the side of a square with the same perimeter as the given rectangle and
GM between a and b is the side of a square with the same area as the given
rectangle with sides a and b. Coefficient n = 2 refers to the use of a square
root in GM. In general on any straight line, an interval [a, b] can be divided
in 2 or more parts following the construction in Figure 1, where the number
of intervals is n = 3 with the relations given by the equations in Figure 1.
The intervals in the left graph show a composition of translations with
itself. In the case of GM the subsequent divisions define homothetic transformations. An interval can be divided in n intervals with n = any natural
number. The second author studied the inverse question: “Can x and y, z...
be determined graphically using only parallel lines?” This is straightforward
in the case of AM but impossible in the case of GM. However, a construction
3
Figure 1: Division of an interval [a, b] into three parts according to AM or
GM.
using only parallel lines yields the harmonic mean HM which is the ratio
between AM and GM (Figure 2).
Figure 2: Graphical construction of the harmonic means HM for n = 3.
This leads to the following relations for the division of an interval in three
parts (Figure 2)
AM 1 HM 2 = ab,
3
3
AM 2 HM 1 = ab,
3
3
GM 1 GM 2 = ab
3
3
(2)
In general:
AM i =
n
AM (n−i)
n
√
n
(n−i)a+ib
nab
, GM i = an−i bi , HM i = ia+(n−i)b
n
n
n
√
n
nab
= ia+(n−i)b
, GM (n−i) = ai bn−i , HM (n−i) = (n−i)a+ib
n
n
n
AM i HM (n−i) = ab,
n
n
AM (n−i) HM i = ab,
n
n
4
GM i GM (n−i) = ab
n
n
(3)
These relationships can be proven using the similarity of triangles, or
from a projective point of view, using double ratios as follows. The double
ratio [a, b, p, q] is harmonic, so [a, b, p, q] = −1. If we take p and q at infinity,
a+b
1 and
=
AM
a,
b,
,
∞
= −1. If we take p = 0 and q at
then x = a+b
2
2
2
2ab
2ab
infinity, then x = a+b = HM 1 and a, b, a+b , 0 = −1 (Figure 3).
2
Figure 3: Projective method for n = 2.
In general the double ratio [a, b, x, y] is calculated as the quotient of the
share ratio’s [a, b, x] and [a, b, y], so [a, b, x, y] = [a,b,x]
. The share ratio [a, b, x]
[a,b,y]
x−a
is calculated by x−b .
If the abscissa of x towards (a, b) is ni then x = AM i for x−a
= ni so
b−a
n
i
x = (n−i)a+ib
and [a, b, x, ∞] = [a, b, x] = x−a
= i−n
= k. If we take now
n
x−b
i−n
1
[a, b, y, 0] = i = k then [a, b, x, ∞] [a, b, y, 0] = 1 and y = HM n−i , for
n
nab
[a, b, y, 0] = i−n
= [a,b,y]
. Since y−a
= ab i−n
, so y = (n−i)a+ib
= HM n−i and
i
[a,b,0]
y−b
i
n
AM i HM n−i = ab.
n
n
This method provides for a recursive method for the calculation of roots
√
[2],[3],[4]. The n−th root n c of a positive number can be interpreted as
√
√
n
GM 1 = 1n−1 c1 of the interval [1, c]. Let x be an approximation of n c,
n
5
√
√
n
c
smaller than n c, then the interval xi , xn−i
includes ci . For this interval
n +ic
ncxn−i
and HM n−i = (n−i)x
AM i = (n−i)x
n−i
n +ic , AM i HM n−i = c. So one could
nx
n
n
n
n
iterate on AM i as well as on HM n−i in:
n
n−i HM n−i =
n
n
n−i
√
ncxn−i
< nc<
n
(n − i)x + ic
i
(n − i)xn + ic
= i AM i (4)
n
nxn−i
√
c
Here x is smaller then n c so we iterate on the interval xi , xn−i
that contains
√
n i
c . If we take i = 1 then we obtain:
√
ncxn−1
(n − 1)xn + c
n
n−1 HM n−1 = n−1
c
<
= AM 1
(5)
<
n
n
(n − 1)xn + c
nxn−1
√
On the right side we recognize the formula of Newton for the zero value n c
of the function f(x) = xn − c. The derivative of f is f (x) = nxn−1 . So with
the tangent method we obtain:
x−
f(x)
(n − 1)xn + c
xn − c
=
=
x
−
f (x)
nxn−1
nxn−1
(6)
This algorithm however, is the shortest of all possible algorithms of this kind.
The speed of convergence is higher with a higher value of i. So the algorithm
on the left side of the last expression is the fastest for the root exponent
(n − 1) has the highest level.
Geometric means and Pascal’s Triangle
It is remarkable that these formulae can be generated with simple geometry
and algebra, represented in beautiful nomograms, without the sophisticated
tools of analysis. It is thus possible to understand various means of different order n geometrically and algebraically. The arguments over which the
nth −root is taken are also the various entries of Pascal’s Triangle wherein
the normal rules of arithmetic are encoded. The coefficients for each term
in the expansion of (a + b)n can be derived using the Binomial theorem of
6
Figure 4: First rows of Pascal’s Triangle.
Newton. Every product between a and b in the Triangle is the argument of
the geometric mean of some order between numbers a and b (Figure 4).
We observe that the order n decreases from n to 0, from left to right for
a and increases for b in the same direction. If we write 1a2 we understand
at the same time that this is equal to b0 a2. In the one direction we have a
lowering of the exponent of a or b, in the other direction an increase according
to the following rules:
(7)
xn → nxn−1
xn → xn+1
(8)
The procedure with Equation 7 is also known as derivation (definition (2)
Symbolic). Performing this in two directions and using proper normalization
dividing by n! the normal binomial coefficients come out.
1a4 → 4a3 → 12a2 → 24a1 → 24
24 ← 24b ← 12b2 ← 4b3 ← 1b4
(9)
Multiplying term by term
24a4
96a3 b
144a2 b2
96ab3
24b4
(10)
Adding all terms and dividing each term by n! = 4!
1a4+4a3b+6a2b2 +4ab3+1b4
7
(11)
Geometrically: A game of cubes and beams
Geometrically, each entry in a given row of Pascal’s Triangle has the same
dimension. The fourth row for example, consists of cubes with side a and b
(and respective volumes a3 and b3, one of each), beams with sides a, a and b
(and volume a2 b, three of them) and beams with sides a, b and b (and volume
ab2, also three of them) [5].
Any row in the Triangle contains pure n−cubes an and bn (Figure 4,
numbers in bold) on the one hand, and n−beams on the other hand (Figure 4,
in grey, non-bold), whose sum is equal to a hypercube (a+b)n . Hypercubes or
n−cubes are n−dimensional cubes (with n > 3, with side a or b), hyperbeams
or n−beams are n−dimensional beams (with n > 3) of which at least one
side is different from all other sides (for example an bm ).
It is in fact also very easy to turn beams like an bm of dimension (n +
m) into cubes of the same dimension. For each hyperbeam an bm one can
construct a hypercube with the same volume by taking the (n + m)−th root
of an bm . This gives the side of the n + m dimensional cube. Which is the
procedure discussed in part 2, on geometric means of a particular order, all
of the same dimension. More specifically for GMi/n between two numbers
√
and b, we specificy i, n, a and b (either a or b can be 1). So, n a = GM 1 of
n
√
√
√
n
n+m
an bm = GM 1 of the
the interval [1, a] since 1n−1 a1 = a and thus
n+m
interval [1, an bm ].
Simon Stevin (1548-1620) reasoned and thought about geometric numbers
(Figure 6). Stevin was one of the greatest mathematicians of the 16th century
and his work was both of a pure and applied nature providing a bridge
between the old and the new sciences [7]. His equilibrium of forces and
the parallelogram rule was the beginning of abstract algebra and of higher
dimensional geometry [4], [6].
In Figure 6 the examples are given of the powers of 2 (upper row) and the
powers of 1 (lower row). 23 is a cube, and 24 (= 16) are two cubes of the size
23 . Likewise 25 (= 32) are 3 of those cubes. For a cube with all sides equal to
1, the results remain the same for any power. It is the neutral element and
8
Figure 5: Simon Stevin of Bruges [7].
any number of multiplications of 1 by itself always yields the same result.
An object like b4 (arithmetically the product b × b × b × b) can be understood geometrically in many different ways. The object b4 is not only a
four-dimensional volume of a hypercube with side b, but it is also b times
a three-dimensional volume b3 (side of this cube is b). At the same time it
can be seen as b × b times an area of b2, but b2 could also be interpreted as
a beam of volume 1 × b2 . This beam can then be made into a cube with
the same volume, but with side (1 × b2 )1/3, the one-third geometric mean
between b2 and 1, or the second geometrical mean of 1 and b, GM2/3 of [1, b]
√
√
√
√
3
3
is 3 13−2 × b2 = b2, GM1/3 of [1, b2 ] is 3 1 × b2 = b2. And so on.
For the four-dimensional volume, it is important to be reminded how
Pascal thought about the fourth dimension: “Et l’on ne doit pas être blessé
par cet quatrième dimension” [8]. Which means that intelligent people should
not be put off by something like the fourth dimension, because in reality it
9
Figure 6: Stevin’s Geometric numbers [5], [6].
is about multiplication.
Stevin’s approach of geometrical numbers was directly related to arithmetic and in Definition XXXI [6] he states that any number can be square,
cube etc., or that also roots are numbers: “Que nombres quelconques peuvent estre nombres quarrez, cubiques etc. Aussie que racigne quelconque est
nombre”. From this Stevin reaches the fundamental conclusion that there
are no absurd, irrational, irregular, inexplicable or surd numbers, “Qu’il ny
a aucuns nombres absurdes, irrationels, irreguliers, inexplicables, ou sourds”
[6].
A contemporary of Stevin, François Viète (1540-1603) prefers to deal
exclusively with numbers avoiding all geometrical connotations. In his “Logistices speciosae canonica praecepta” (canonical rules of species calculation
[9]), the main law (Lex homogeneorum) states that only species of the same
kind (homogeneous species) can be added or subtracted. In a typical row of
Pascal’s Triangle, all n−cubes and n−beams are of the same dimension or
the same species (speciosa), but in general polynomials (for example in one
variable) this is not the case and here the Lex homogeneorum rules, according
10
to “common sense”. This “in fact-not-so-common-sense” is one of the main
reasons for the split between arithmetic and geometry.
Contrary to Viète’s Lex homogeneorum however, it is very easy to get the
same dimension for any term of a polynomial using the unit element. For
example, a polynomial like x4 + x3 + x can be written as (x × x × x × x) +
(x × x × x × 1) + (x × 1 × 1 × 1); all of the same dimension and actually, all
geometric means of different orders between x and the unit element 1 can be
written this way [4]
3
3
3
3
2
3
2
(1 × x ) + 3 (12 × x)
(12)
x +x +x=x +
It is of interest to add here another definition of Stevin in his first book on
arithmetic, namely definition XXVI “Multinomie algebraique est un nombre
consistent de plusieurs diverses quantitez”. This definition introduces the
reader to algebraic multinomials or polynomials, “Comme 3z + 5y − 4x + 6
s’appele multinome algebraique. Et quand il aura de quantitez comme 2x+4y
s’appelent binomie, et de trois quantitez s’appellera trinomie, etc.” [6].
The decimal principle and fluxions
All this was very natural for mathematicians like Simon Stevin and his contemporaries. René Descartes wrote: “Just as the symbol c1/3 is used to represent the side of a cube a3 has the same dimension as a2b” [10]. The relation
between products and rectangles was frequently used for didactical reasons,
for example by John Colson in his “perpetual comment” to Newton’s Method
of fluxions and infinite series - To which is subjoin’d: A perpetual comment
upon the whole work, consisting of annotations, illustrations and supplements
in order to make this treatise A Compleat Institution for the use of learners
[11].
This treatise brings out a nice historical connection between Stevin and
Newton, going back to the complete arithmetical treatment with natural
numbers by Stevin in the decimal system in 1585. Stevin showed in his
book De Thiende that a complete arithmetical control of the real number
11
system is achieved by explicitly demonstrating how all operations on and
with real numbers can be carried out when expressing these numbers in the
decimal system. Stevin added in an appendix to De Thiende [12] that the
decimal principle should be advocated in “all human accounts and measurements”, thereby “anticipating the (partial) realization of this simple idea by
two centuries”. The importance of the decimal principle for geometry and
for science cannot be overemphasized. It opened the way to Descartes algebraic geometry and inspired Newton to write his Method of fluxions and
infinite series with its application to the geometry of curve-line [11] from the
following motivation:
“Since there is a great conformity between the Operations in Species,
and the same Operations in common Numbers; nor do they seem
to differ, except in the Characters by which they are presented,
the first being general and indefinite, and the other definite and
particular: I cannot but wonder that no body has thought of accommodating the lately-discover’d Doctrine of Decimal Fractions
in like manner to Species,..., especially since it might have open’d
a way to more abstruse Discoveries. But since this Doctrine of
Species, has the same relation to Algebra, as the Doctrine of Decimal Numbers has to common Arithmetick: the Operations of Additions, Subtractions, Multiplication, Division and the Extraction
of Roots, may easily be learned from thence, if the Learner be but
skilled in Decimal Arithmetick, and the Vulgar Algebra, and observes the correspondence that obtains between Decimal Fractions
and Algebraick Terms infinitely continued. For as in Numbers,
the Places towards the right-hand continually decrease in a Decimal or Subdecuple Proportion; so it is in Species respectively,
when the Terms are disposed in an uniform Progression infinitely
continued, according to the Order of the Dimensions of any Numerator or Denominator. And as the convenience of Decimals
is this, that all vulgar Franctions and Radicals, being reduced to
12
them, in some measure acquire the nature of Integers, and may be
managed as such, so it is a convenience attending infinite Series
in species, that all kinds of complicate Terms may be reduced to
the Class of simple Quantities...”
One of the major development in the book concerns infinite series. John
Colson (1680-1760) who later became Lucasian professor of Mathematics, as
one of the successors of Barrow and Newton, wrote in his Introduction [11]:
“As to the Method of Infinite Series, in this the Author opens
a new kind of Arithmetick, (new at least at the time of writing
this), or rather he vastly improves the old. For he extends the
received Notation, making it completely universal and shews, that
as our common Arithmetick of Integers received a great improvement by the introduction of decimal Fractions; so the common
Algebra or Analyticks, as an universal Arithmetick, will receive
a like Improvement by the admission of his Doctrine of Infinite
Series, by which the same analogy will be still carry’d on, and
farther advanced towards perfection. Then he shews how all complicate Algebraical Expressions may be reduced to such Series, as
will continually converge to the true values of those complex quantities or their Roots, and may be therefor be used in their stead.”
In Taylor and MacLaurin series, the same rules as above (Equations 7,
D(xn ) = nxn−1 ) are key. They give an operational definition of functions
with for example, the MacLaurin series for ex , cosine and sine. This is
shown in Figure 7 for sine.
2
3
4
ex = 1 + 1!x + x2! + x3! + x4! + ...
3
5
7
sin x = x − x3! + x5! − x7! + ...
2
4
6
cos x = 1 − x2! + x4! − x6! + ...
(13)
A key to understand derivatives is putting the unit element back where it
belongs.
13
Figure 7: Sine function for increasing number of terms in partial sums.
Indeed, performing derivation is substituting x by 1, one at the time in
a series. In Equation 6 in the process of derivation (e.g. of x3 ) we observe
x3 = (xxx) → 3(xx1) or 1 cube of x3 is compared to beams of sides x, x and
1, and we need three of them.
So, if we want to find the derivative of ex we substitute in every term of
the series one factor x by 1 (and if there is no x as in the first term it becomes
0; the “lowering of the exponent operation”) and put the original exponent
up front lowering n! to (n − 1)!. The result is the same series, as expected.
It is easy to show that the same occurs if we use the series expansions for
sine and cosine, based on the relationship of Euler, D (sin x) = cos x and
D (cos x) = −sin x.
In general, derivatives are related to (higher order) geometric means between two numbers f and e. Higher order geometric means between two
pure numbers f and e involve expressions of the type (m + n)−th root
of f m × en (Figure 8). For e = 1, f 3 + f 2 + f + 1 can be written as
f × f × f + f × f × e + f × e × e + e × e × e (Figure 8).
14
Figure 8: From Newton’s Principia [13].
The geometry of parabolas
The use of the unit element allows for comparing n−volumes (converting m
volumes into n−volumes of the ‘same’ dimension if needed). This, in our
view, also shows that the Greeks were well aware of “units” in relation to
conic sections. In a parabola (y = x2), the variable y scales to the first power
while some other variable x scales to the second, but geometrically a parabola
indicates that for each coordinate x, one can construct (in Greek terminology
to each line with length x it is possible to apply) a square with area equal
to x2 , such that this area corresponds exactly to the area of a rectangle with
width 1 and height y (x × x = y × 1). In this sense the parabola is an
“equiareal” figure. This was known to Ancient Greek geometers and is at
the basis of the conic sections.
Allometric equations and power laws, expressing the constancy of relative
growth and generally depicted as straight lines in log-log plots, can be understood in the same geometric way [5], namely that these equations express
some conservation law for n−volumes of n−cubes and n−beams, with the
parabola and hyperbola for n = 2. The power law y = x3/4 or equivalently
y 4 = x3 thus states that the 4−cube with side y is exactly the same as the
4−beam with four sides x, x, x and 1 (with volume x × x × x × 1). In our
days y = x3/4 is related often to “fractal dimension”, but these are parabola’s
of the type y n = xn−1 (Eq. 6). In a general way the parabolas of the family
y n = nxn−1 are nothing but derivatives. Reading the graphs the other way,
y n /n = xn−1 stands for integration. Simple and pure.
An example from physics is Kepler’s Law of Periods, which states that
the square of the orbital period of a planet is directly proportional to the
cube of the semi-major axis of the elliptical orbit. So the volume of a beam
15
formed by the orbital period T1 and the unit element is T1 × T1 × 1 (i.e. the
volume of a beam with height and length T1 and width 1), equals the volume
of a cube a1 × a1 × a1 (with sides equal to the semi-major axis a1 ) up to
a constant. On a weighing balance this constant can be interpreted as the
length of the arms. This is also an equi-areal law.
T1
2
4π 2
a13
=
G(M1 + M2 )
(14)
This in our opinion at least, settles the question whether the Greeks understood derivatives and dynamics in an affirmative way.
The examples of parabola and ellipse show that they understood the
unit element in a general way, not necessarily as a number. It was only in
the Renaissance that it really became a number. Simon Stevin, was one of
the first to state that the unit was a number “Que l’Unité est Nombre”. It
becomes the neutral element for multiplication, but it certainly does not need
to be “one”, as long as the element selected is “neutral”. If we look back at
Figure 1 and move the point zero as far as possible to the left, and define a
neutral element close to zero, this element becomes as far as possible removed
from a and b, having the least possible influence for computing AM or GM.
One can call such an infinitely small number for example “dx”. Obviously,
one does not need to move zero; easier is to move the neutral element as close
as possible to zero.
The ‘rectangles’ xdx in Riemann sums for integrals are a generalization of
parabolas, but viewed from a Greek’s geometrical perspective, it does not add
anything really new. The question whether dx = 0 or 1 becomes then pretty
irrelevant. It needs to be neutral in multiplication (in which case dx = 1),
and if it needs to be neutral for addition (x + dx) it has the tendency to be
rather close to zero or any number close enough.
Going back to Figure 1 and considering GM and AM (for n = 2) one
understands that the geometric mean (GM) is strictly smaller than the arithmetic mean (AM). In order to have AM approach GM, the point 0 where
the lines cross has to be moved as far to the left as possible. Equality is
16
obtained only when lines run parallel. This is an arithmetical interpretation
of Euclid’s fifth postulate and an illustration of Shiing-Shin Chern’s remark
that “Euclid’s Elements are a geometrical treatment of the number system”
[14]: the fifth postulate ensures that AM and GM can be constructed and
that, for any two numbers a and b on a line, GM is strictly smaller than
AM, which is the cornerstone of our number system [5].
1, 2,...,11,...,37: monomiality principle for polynomials
The explicit forms of the series expansions for the exponential, sine and cosine
functions are:
x
∞
e =
n=0
1n n
x
n!
∞
sin x =
n=0
(−1)n 2n+1
x
(2n + 1)!
∞
cos x =
n=0
(−1)n 2n
x
(2n)!
(15)
Sine and cosine are examples of simple polynomials, but in mathematics there
are a large number of special polynomials in the theory of special functions,
such as Legendre, Hermite, Laguerre, Bell, Appell... polynomials (Figure
9). Special functions play an important role in various field of mathematics,
physics and engineering and during the last decades new families of special
functions have been suggested in various branches of physics [15].
Comparing many such special functions and their explicit form in contemporary pure and applied mathematics to functions of a more elementary
nature (ex, sine, cosine...) from the 17th and 18th century seems somewhat like jumping from older, simpler definitions of the derivative (list of
Thurston 1 through 7) to definition 37. Indeed, at first sight contemporary
special polynomials are a far cry from the simpler functions and power series,
from parabolas and cubes and beams. But is that so?
Most families of special polynomials can be transformed by the same
simple procedure of raising and lowering exponents, the same game of means
and beams. By virtue of the so-called monomiality principle, all families
of polynomials, and in particular special polynomials, can be obtained by
transforming a basic monomial set by means of suitable operators P and M,
17
Figure 9: Example of Hermite polynomials.
called the derivative and multiplication operator of the considered family,
respectively.
The definition of the poweroid introduced by J. F. Steffensen has been
framed in the monomiality principle by G. Dattoli [16], providing a very
powerful analytical tool for deriving properties of special polynomials, such as
Hermite, Bessel, Laguerre, Bell and Legendre polynomials [17],[18],[19],[20].
Let us consider the Heisenberg-Weyl algebra with generators P and M
satisfying the commutation relation [P, M] = P M − MP = 1, and the family
of polynomials π = pn (x) (n = 0, 1, 2, ...). Then, π is quasi-monomial if the
identities P (pn (x)) = npn−1 (x) and M (pn (x)) = pn+1 (x) hold true. In
this case, the analytical properties of π can be obtained straightforwardly
starting from those of the operators P and M. As an example, combining
the mentioned identities yields immediately the governing equation of the
general polynomial pn (x) belonging to π, namely (MP − n) (pn (x)) = 0.
Furthermore, it is worth noting that, under the assumption that p0 (x) = 1,
pn (x) can be represented explicitly as pn (x) = M n (1).
The simplest application of the monomiality principle regards the family
18
of monomials {xn } in one variable with derivative and multiplication operators P = D = d/dx and M = x respectively, such that [P, M] = 1 and
M (xn ) = xn+1 , P (xn ) = nxn−1 . These operations simply consist in raising
and lowering the exponents of the general monomial belonging to the family.
Additional relations are given by x0 = 1 and Dx0 = 0 (One can easily see
that it is also possible to start from the antiderivative).
Using the monomiality principle, the analytical properties of special polynomials can be easily studied. Following this approach, the governing differential equation, recurrence relations and identities can be easily determined.
As an example, let us consider the family of Hermite polynomials in two
variables [21]:
[ mn ]
n!
(16)
y k xn−km
Hn(m) (x, y) :=
k!(n
−
km)!
k=0
(m)
with Hn (x, 0) := xn and m = 1, 2, . . . , n. As shown in [20] for the case
m = 2, these polynomials are particular solutions of the generalized two(m)
(m)
dimensional heat equation ∂y Hn (x, y) = ∂xm Hn (x, y).
Therefore, the general polynomial can be represented in a very com(m)
n
pact way as Hn (x, y) = (x + my∂x) (1), whereas the relevant govern(m)
(m)
ing differential
equation is obtained as my∂x2Hn (x, y) + x∂xHn (x, y) =
(m)
(m)
MP Hn (x, y) = nHn (x, y). Other recurrence formulae and identities
(m)
involving Hn (x, y) for m = 2 are derived in a similar way in [20], which
provides an excellent overview on the application of operational techniques to
special polynomials. Similar unifying approaches can be developed starting
from the Pascal matrix [21].
-1, -2, -3, ...: understand the legacy
For Thurston, the closest definition of mathematics is “the theory of formal
patterns” and “mathematicians are those humans who advance human understanding of mathematics” [1]. In this article we showed that the formal
pattern underlying various means, the normal rules of arithmetic, expansions
19
and special functions is in principle a game of cubes and beams, going back
to Greek foundations (-1, -2, -3 is counting down to some time, say -2500
years, when Pythagoreans mastered means, numbers and much more). With
simple principles and elementary functions we can achieve already a very
good understanding of mathematics and the relations among various fields.
We hope that this article, with its didactic emphasis, can contribute to a
better human understanding of these topics.
Newton’s intuition (“if the Learner be but skilled in Decimal Arithmetick,
and the Vulgar Algebra...”), is still accurate today, but incomplete. Basic
geometry is also required for a better and more flexible understanding. To
understand higher dimensional cubes and beams as simple geometric numbers, cubes, beams and balances in equilibrium, lift the ban imposed by Viète
[9] in the 17th century and by Grassman/Peano [22] in the 19th century. To
this very day this ban continues to exist. Even Stephen Hawking [10] notes
on La Géometrie of Descartes: “At the time this was written a2 was commonly considered to mean the surface of a square whose side is a, and b3 to
mean the volume of a cube whose side is b3 while b4 , b5... were unintelligible
as geometric forms”.
René Descartes (1596-1650) certainly knew about the works of Simon
Stevin (1548-1620), not only because they were available in French, but also
because of his collaboration with Isaac Beeckman, himself a student of Stevin.
Anyway, two centuries later Monge, Gauss, Lamé and Riemann among others
brought geometry back onto center stage [23].
With regard to the intimate relation of algebra and geometry and the
various idle discussions on which of these two is more important, consider
this end-of-20th-century attempt to define special functions [15]:
“It is also difficult to frame in an univocal way the concept of
Special Function itself. Just to make an attempt, we can associate Special Functions with the solutions of particular families
of ordinary differential equations with non-constant coefficients.
During the end of the last century, Sophus Lie pondering on the
20
deep reasons underlying the solution by quadrature of differential
equations was led to the notion of group symmetry. This concept
inspired the work of Cartan, who was the first to point out that
Special Functions can be framed within the context of the Lie theory. This point of view culminated in the work of Wigner who
regarded the Special Functions as matrix elements of irreducible
representations of Lie groups.”
This is not the most general definition, since it leaves out some polynomials,
but this quote is intended to illustrate the deep connections among fields.
These very fundamental relations where summarized by Chern in the following: “While algebra and analysis provide the foundations of mathematics,
geometry is at the core” [14]. This trinity is reflected in this article, with one
underlying principle, understood very well in ancient Greece.
Greek and Hellenistic mathematics and science was advanced in every
sense of the word [24]. The rebirth of Eudoxus mathematical findings into
Dedekind’s cut is one example. We referred earlier to Chern’s statement
that Euclid’s Elements are a geometrical treatment of the number system.
The influence of Stevin’s Decimal fractions on the development of fluxions
has been pointed out, but this was really based on this very same idea of
commensurability. As D. J. Struik (editor of the mathematical parts of the
Principal Works of Simon Stevin) writes:
“In his arithmetical and geometrical studies, Stevin pointed out
that the analogy between numbers and line-segments was closer
than was generally recognized. He showed that the principle arithmetical operations, as well as the theory of proportions and the
rule of three, had their counterparts in geometry. Incommensurability existed between line-segments as well as numbers...; incommensurability was a relative property, and there was no sense
in calling numbers “irrational”, “irregular”, or any other name,
which connoted inferiority. He went so far as to say, in his
21
Traicté des incommensurable grandeurs, that the geometrical theory of incommensurables, in Euclid’s Tenth Book had originally
been discovered in terms of numbers, and translated the content
of this book into the language of numbers. He compared the still
incompletely understood arithmetical continuum to the geometrical continuum already explained by the Greeks, and thus prepared
the way for that correspondence of numbers and points on the line
that made its entry with Descartes’ coordinate geometry [6].”
In some sense, all we are doing is adding some commas and punctuations
here and there to what the Greek have done. Also on the historical side
much is to be learned from Bacon’s writings [25]:
“So that as Plato had an imagination that all knowledge was but
remembrance; so Solomon giveth his sentence, that all novelty is
but oblivion.”
One example of such oblivion and “novelty” are the series expansions for cosine and sine, as well as their definitions, traditionally associated with names
of 16th and 17th century Western mathematicians. In fact, trigonometric
functions sine and cosine as well as their expansion were known already in
the early 15th century to the Indian mathematician Madhava of Sangamagramma (c.1340-1425). Pascal’s Triangle was not invented by Pascal, but
was known long before in China and Persia. Many special polynomials were
invented a century ago and then forgotten, until rediscovered by physics [15].
The history of mathematics is very old and its legacy extremely rich, transcending boundaries in space and time.
On landscapes and maps
Another goal of this article, other than pointing out that all these things are
intrinsically linked spanning a period of at least 2,5 millennia, is to show how
important it is to understand concepts in different ways. Thurston’s list is “a
22
list of different ways of thinking about or conceiving of the derivative, rather
than a list of different logical definitions” [1]. It is important to be able to
look at concepts, for which mathematicians spent thousands of years to come
to an ever-better understanding, from very different perspectives, not only
tailored to specific talents of individuals, but to safeguard the true spirit of
mathematics. Thurston again: “Unless great efforts are made to maintain
the tone and flavor of the original human insights, the differences start to
evaporate as soon as the mental concepts are translated into precise, formal
and explicit definitions” [1]. One example is the development of algebra and
its deviation from geometric numbers.
One of the defining characteristics of Greek mathematics was not only the
development of mathematics, but also to understand that this has an intimate
relationship to the workings of the world [24]. According to Feynman [26] “To
those who do not know mathematics it is difficult to get across a real feeling
as to the beauty, the deepest beauty, of nature... If you want to learn about
nature, to appreciate nature, it is necessary to understand the language that
she speaks in. She offers her information only in one form”. Fortunately,
simple rules are a basic feature of this language, by whatever name they
are known, addition and multiplication, means, cubes or the monomiality
principle [18].
Despite the simple basic rules from which a wide range of methods can
be deduced, contemporary mathematics has evolved into very diverse landscapes, in pure and applied mathematics or in theoretical physics. Each uses
one or more very specific high-level (increasingly abstract) languages, with
various local dialects. A mathematical concept like curvature goes under
a variety of different names in mathematics and physics, dependent on the
field. There is the imminent danger of high-level languages for different landscapes (the languages as the maps or Des Cartes), taking precedence over
these landscapes and territories themselves.
“The transfer of understanding from one person to another is
not automatic. It is hard and tricky. Therefore, to analyze hu23
man understanding of mathematics, it is important to consider
who understands what, and when. Much of the difficulty has to
do with the language and culture of mathematics, which is divided
into subfields. Basic concepts used every day within one subfield
are often foreign to another subfield. Mathematicians give up
on trying to understand the basic concepts even from neighboring subfields, unless they were clued in as graduate students. In
contrast, communication works very well within the subfields of
mathematics. Within a subfield, people develop a body of common
knowledge and known techniques [1].”
When it can be shown that simple rules underlie different mathematical
landscapes, the general language (the art of mapmaking) need not be too
abstract for a basic understanding or the development of a certain feeling for
the matter.
The “maps” we spoke of are based on the Pythagorean theorem or triangles in general (including triangulation of surfaces), and geometric means
between numbers. There are other ways to make maps to better understand
nature. A generalization of the Pythagorean Theorem based on n-cubes (instead of squares) leads to the simplest cases of Minkowski-Finsler geometry
and the curves associated with this generalization are Lamé curves, named
after Gabriel Lamé (1795-1870). A subclass of Lamé curves are superellipses
[27], defined by:
x n y n
(17)
+ =1
A
B
This in fact considers cubes (and n−cubes) only, not using beams or geometric means between numbers and hence the relation of Lamé curves to
the Last Theorem of Fermat. The coefficients of the expansion of (2x + 1)n
have a nice geometrical meaning [27, Chapter 4] for n = 4, (2x + 1)4 =
16x4 + 32x3 + 24x2 + 8x + 1, a tesseract (a four dimensional cube) is composed of 16 points, 32 lines, 24 squares, 8 cubes and of course 1 tesseract.
Again, this is a nice example of lowering of exponent with a clear geometrical
meaning.
24
A recent generalization of Lamé curves provides a new way of studying geometry and natural shapes [28],[29],[30] (Figure 10 for some natural
shapes), whereby Geometry becomes intimately connected with Growth and
Form in nature [31]. A geometric study of natural shapes and phenomena can
be proposed using only pure numbers an or variables xn without geometric
means.
Figure 10: Some natural shapes as transformations of a circle [27], [28].
As René Thom stated: “Geometry is successful magic!” [32]1 . Simon
Stevin’s motto was “Wonder is no Wonder” [33], meaning that geometrical
understanding is the way to get rid of miracles and wonders2 . This remains
true to this very day and in the future. André Weil [34] wrote:
“Obviously everything in differential geometry can be translated
into the language of analysis, just as everything in algebraic geometry can be expressed in the language of algebra. Whether one
considers analytic geometry in the hands of Lagrange, tensor calculus at this of Ricci, or more modern examples, it is always
clear that a purely formal treatment of geometric topics would invariably have killed the subject if it had not been rescued by true
1
“La Géometrie est magie qui réussit”. Geometry is magic that works, successfully.
The English translation of the biography of Simon Stevin by Van den Berghe and
Devreese is “Miracle is no miracle” [33]. Personally I prefer wonder over miracle. Wonder
in Dutch occurs in the words wonderlijk (strange, odd, surprising), verwondering (wonder, astonishment, surprise). Wonder has a much broader meaning and is therefore less
miraculous than miracle (miracles would be a main thesis of Spinoza almost one century
later, based on a Newtonian laws and, indirectly, Stevin’s Motto.
2
25
geometers, Monge in one instance, Levi-Civita, and above all Elie
Cartan in another... The psychological nature of true geometrical
intuition will perhaps never be cleared up... Whatever the truth
of the matter, mathematics in our century would not have made
such impressive progress without the geometric sense of Elie Cartan, Heinz Hopf, Chern and a very few more. It seems safe to
predict that such men will always be needed if mathematics is to
go on as before.”
Geometry improves not only the understanding of mathematics, but also
connects with the foundations. Radu Miron wrote:
“If Mathematics could be torn from its foundations, it would become a series of formulae, recipees and tautologies that could not
be applied any longer to the objective reality, but only to some
rigid, mortified scheme of this reality.” [35]
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