In
mathematics, the blancmange curve is a
self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after
Teiji Takagi
Teiji Takagi (高木 貞治 ''Takagi Teiji'', April 21, 1875 – February 28, 1960) was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. The Blancmange curve, the graph of a nowhere-differentiabl ...
who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the curve named after Takagi and
Georg Landsberg
Georg Landsberg (January 30, 1865 – September 14, 1912) was a German mathematician, known for his work in the theory of algebraic functions and on the Riemann–Roch theorem.. The Takagi–Landsberg curve, a fractal that is the graph of a nowh ...
. The name ''blancmange'' comes from its resemblance to a
Blancmange pudding. It is a special case of the more general
de Rham curve
In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.
The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all spe ...
; see also
fractal curve
A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectif ...
.
Definition
The blancmange function is defined on the
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
by
:
where
is the
triangle wave
A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.
Like a square wave, the triangle wave contains only odd harmonics. However, ...
, defined by
,
that is,
is the distance from ''x'' to the nearest
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
.
The Takagi–Landsberg curve is a slight generalization, given by
:
for a parameter
; thus the blancmange curve is the case
. The value
is known as the
Hurst parameter
The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases.
Studies involving the Hurst expon ...
.
The function can be extended to all of the real line: applying the definition given above shows that the function repeats on each unit interval.
The function could also be defined by the series in the section
Fourier series expansion.
Functional equation definition
The periodic version of the Takagi curve can also be defined as the ''unique bounded solution
to the functional equation''
:
.
Indeed, the blancmange function
is certainly bounded, and solves the functional equation, since
:
.
Conversely, if
is a bounded solution of the functional equation, iterating the equality one has for any ''N''
:
, for
whence
. Incidentally, the above functional equations possesses infinitely many continuous, non-bounded solutions, e.g.
Graphical construction
The blancmange curve can be visually built up out of triangle wave functions if the infinite sum is approximated by finite sums of the first few terms. In the illustrations below, progressively finer triangle functions (shown in red) are added to the curve at each stage.
Properties
Convergence and continuity
The infinite sum defining
converges absolutely
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
for all
: since
for all
, we have:
:
if
.
Therefore, the Takagi curve of parameter
is defined on the unit interval (or
) if
.
The Takagi function of parameter
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
. Indeed, the functions
defined by the partial sums
are continuous and
converge uniformly toward
, since:
:
for all x when
.
This value can be made as small as we want by selecting a big enough value of ''n''. Therefore, by the
uniform limit theorem
In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous.
Statement
More precisely, let ''X'' be a topological space, let ''Y'' be a metric space, and let ƒ''n'' :&n ...
,
is continuous if , ''w'', <1.
File:Blancmange k1.5.png, parameter w=2/3
File:Blancmange k2.png, parameter w=1/2
File:Blancmange k3.png, parameter w=1/3
File:Blancmange k4.png, parameter w=1/4
File:Blancmange k8.png, parameter w=1/8
Subadditivity
Since the absolute value is a
subadditive function In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element ...
so is the function
, and its dilations
; since positive linear combinations and point-wise limits of subadditive functions are subadditive, the Takagi function is subadditive for any value of the parameter
.
The special case of the parabola
For
, one obtains the
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
: the construction of the parabola by midpoint subdivision was described by
Archimedes.
Differentiability
For values of the parameter
the Takagi function
is differentiable in classical sense at any
which is not a
dyadic rational. Precisely,
by derivation under the sign of series, for any non dyadic rational
one finds
:
where
is the sequence of binary digits in the
base 2 expansion of
, that is,
. Moreover, for these values of
the function
is
Lipschitz of constant
. In particular for the special value
one finds, for any non dyadic rational
*
*
* Benoit Mandelbrot, "Fractal Landscapes without creases and with rivers", appearing in ''The Science of Fractal Images'', ed. Heinz-Otto Peitgen, Dietmar Saupe; Springer-Verlag (1988) pp 243–260.
* Linas Vepstas,
', (2004)
* Donald Knuth, The Art of Computer Programming, volume 4a. Combinatorial algorithms, part 1. . See pages 372–375.