{"history":[],"article":{"id":"3d40a185-4cee-4d0f-9584-352580bec1bc","title":"Blancmange curve","slug":"blancmange-curve","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **blancmange curve** is a [self-affine](https://bliptext.com/articles/self-affinity) [fractal curve](https://bliptext.com/articles/fractal-curve) constructible by midpoint subdivision.\nIt is also known as the **Takagi curve**, after [Teiji Takagi](https://bliptext.com/articles/teiji-takagi) who described it in 1901, or as the **Takagi–Landsberg curve**, a generalization of the curve named after Takagi and [Georg Landsberg](https://bliptext.com/articles/georg-landsberg).\nThe name blancmange comes from its resemblance to a [Blancmange pudding](https://bliptext.com/articles/blancmange).\nIt is a special case of the more general [de Rham curve](https://bliptext.com/articles/de-rham-curve).\n# [A plot of the blancmange curve. | upload.wikimedia.org/wikipedia/commons/9/9a/Blancmange.svg]\n\n# Definition\nThe blancmange function is defined on the [unit interval](https://bliptext.com/articles/unit-interval) by\nwhere s(x) is the [triangle wave](https://bliptext.com/articles/triangle-wave), defined by ,\nthat is, s(x) is the distance from x to the nearest [integer](https://bliptext.com/articles/integer).\nThe Takagi–Landsberg curve is a slight generalization, given by\nfor a parameter w; thus the blancmange curve is the case w=1/2.\nThe value H=-\\log_2 w is known as the [Hurst parameter](https://bliptext.com/articles/hurst-parameter).\nThe function can be extended to all of the real line: applying the definition given above shows that the function repeats on each unit interval.\n\n## Functional equation definition\nThe periodic version of the Takagi curve can also be defined as the unique bounded solution to the functional equation\nIndeed, the blancmange function T_w is certainly bounded, and solves the functional equation, since\nConversely, if T:\\R\\to\\R is a bounded solution of the functional equation, iterating the equality one has for any N\nwhence T=T_w.\nIncidentally, the above functional equations possesses infinitely many continuous, non-bounded solutions, e.g.\n\n## Graphical construction\nThe 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.\nIn the illustrations below, progressively finer triangle functions (shown in red) are added to the curve at each stage.\n\n# Properties\n\n## Convergence and continuity\nThe infinite sum defining T_w(x) [converges absolutely](https://bliptext.com/articles/absolute-convergence) for all x. Since for all\nif |w|<1.\nThe Takagi curve of parameter w is defined on the unit interval (or \\mathbb{R}) if |w|<1.\nThe Takagi function of parameter w is [continuous](https://bliptext.com/articles/continuous-function).\nThe functions T_{w,n} defined by the partial sums\nare continuous and [converge uniformly](https://bliptext.com/articles/uniform-convergence) toward T_w:\nfor all x when |w| < 1.\nThis bound decreases as n\\to\\infty.\nBy the [uniform limit theorem](https://bliptext.com/articles/uniform-limit-theorem), T_w is continuous if |w| < 1.\n\n## Subadditivity\nSince the absolute value is a [subadditive function](https://bliptext.com/articles/subadditive-function) so is the function, and its dilations s(2^kx); since positive linear combinations and point-wise limits of [subadditive function](https://bliptext.com/articles/subadditive-function)s are subadditive, the Takagi function is subadditive for any value of the parameter w.\n\n## The special case of the parabola\nFor w=1/4, one obtains the [parabola](https://bliptext.com/articles/parabola): the construction of the [parabola](https://bliptext.com/articles/parabola) by midpoint subdivision was described by [Archimedes](https://bliptext.com/articles/archimedes).\n\n## Differentiability\nFor values of the parameter 0< w < 1/2, the Takagi function T_w is differentiable in the classical sense at any x\\in\\R which is not a [dyadic rational](https://bliptext.com/articles/dyadic-rational).\nBy derivation under the sign of series, for any non dyadic rational x\\in\\R, one finds\nwhere is the sequence of binary digits in the [base 2](https://bliptext.com/articles/base-2) expansion of x:\nEquivalently, the bits in the binary expansion can be understood as a sequence of [square waves](https://bliptext.com/articles/square-wave), the [Haar wavelets](https://bliptext.com/articles/haar-wavelet), scaled to width 2^{-n}.\nThis follows, since the derivative of the triangle wave is just the square wave:\nand so\nFor the parameter 0< w < 1/2, the function T_w is [Lipschitz](https://bliptext.com/articles/lipschitz-continuity) of constant 1/(1-2w).\nIn particular for the special value w=1/4 one finds, for any non dyadic rational x\\in[0,1], according with the mentioned\nFor w=1/2 the blancmange function T_w it is of [bounded variation](https://bliptext.com/articles/bounded-variation) on no non-empty open set; it is not even locally Lipschitz, but it is quasi-Lipschitz, indeed, it admits the function as a [modulus of continuity](https://bliptext.com/articles/modulus-of-continuity).\n\n## Fourier series expansion\nThe Takagi–Landsberg function admits an absolutely convergent Fourier series expansion:\nwith and, for m\\ge 1\nwhere 2^{\\nu(m)} is the maximum power of 2 that divides m. Indeed, the above [triangle wave](https://bliptext.com/articles/triangle-wave) s(x) has an absolutely convergent Fourier series expansion\nBy absolute convergence, one can reorder the corresponding double series for T_w(x):\nputting m=2^n(2k+1) yields the above Fourier series for T_w(x).\n\n## Self similarity\nThe [recursive definition](https://bliptext.com/articles/) allows the [monoid](https://bliptext.com/articles/monoid) of self-symmetries of the curve to be given.\nThis monoid is given by two generators, g and r, which [act](https://bliptext.com/articles/group-action-mathematics) on the curve (restricted to the unit interval) as\nand\nA general element of the monoid then has the form for some integers This [acts](https://bliptext.com/articles/group-action-mathematics) on the curve as a [linear function](https://bliptext.com/articles/linear-function): for some constants a, b and c.\nBecause the action is linear, it can be described in terms of a [vector space](https://bliptext.com/articles/vector-space), with the [vector space](https://bliptext.com/articles/vector-space) basis:\nIn this [representation](https://bliptext.com/articles/group-representation), the action of g and r are given by\nand\nThat is, the action of a general element \\gamma maps the blancmange curve on the unit interval [0,1] to a sub-interval for some integers m, n, p.\nThe mapping is given exactly by where the values of a, b and c can be obtained directly by multiplying out the above matrices.\nThat is:\nNote that is immediate.\nThe monoid generated by g and r is sometimes called the **dyadic monoid**; it is a sub-monoid of the [modular group](https://bliptext.com/articles/modular-group).\nWhen discussing the modular group, the more common notation for g and r is T and S, but that notation conflicts with the symbols used here.\nThe above three-dimensional representation is just one of many representations it can have; it shows that the blancmange curve is one possible realization of the action.\nThat is, there are representations for any dimension, not just 3; some of these give the [de Rham curves](https://bliptext.com/articles/de-rham-curve).\n\n# Integrating the Blancmange curve\nGiven that the [integral](https://bliptext.com/articles/integral) of from 0 to 1 is 1/2, the identity allows the [integral](https://bliptext.com/articles/integral) over any interval to be computed by the following relation.\nThe computation is recursive with computing time on the order of log of the accuracy required.\nDefining\none has that\nThe [definite integral](https://bliptext.com/articles/definite-integral) is given by:\nA more general expression can be obtained by defining\nwhich, combined with the series representation, gives\nNote that\nThis integral is also self-similar on the unit interval, under an action of the dyadic monoid described in the section [Self similarity](https://bliptext.com/articles/).\nHere, the representation is 4-dimensional, having the basis.\nThe action of g on the unit interval is the [commuting diagram](https://bliptext.com/articles/commuting-diagram)\nFrom this, one can then immediately read off the [generators](https://bliptext.com/articles/generating-set-of-a-group) of the four-dimensional representation:\nand\nRepeated integrals transform under a 5,6,... dimensional representation.\n\n# Relation to simplicial complexes\nLet\nDefine the Kruskal–Katona function\nThe [Kruskal–Katona theorem](https://bliptext.com/articles/kruskal-katona-theorem) states that this is the minimum number of (t − 1)-simplexes that are faces of a set of N t-simplexes.\nAs t and N approach infinity,\n(suitably normalized) approaches the blancmange curve.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T07:16:32.200Z","updated_at":"2025-01-26T07:16:32.200Z","search_vector":"'blancmang':1 'curv':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **blancmange curve** is a [self-affine](https://bliptext.com/articles/self-affinity) [fractal curve](https://bliptext.com/articles/fractal-curve) constructible by midpoint subdivision.\nIt is also known as the **Takagi curve**, after [Teiji Takagi](https://bliptext.com/articles/teiji-takagi) who described it in 1901, or as the **Takagi–Landsberg curve**, a generalization of the curve named after Takagi and [Georg Landsberg](https://bliptext.com/articles/georg-landsberg).\nThe name blancmange comes from its resemblance to a [Blancmange pudding](https://bliptext.com/articles/blancmange).\nIt is a special case of the more general [de Rham curve](https://bliptext.com/articles/de-rham-curve).\n# [A plot of the blancmange curve. | upload.wikimedia.org/wikipedia/commons/9/9a/Blancmange.svg]\n\n# Definition\nThe blancmange function is defined on the [unit interval](https://bliptext.com/articles/unit-interval) by\nwhere s(x) is the [triangle wave](https://bliptext.com/articles/triangle-wave), defined by ,\nthat is, s(x) is the distance from x to the nearest [integer](https://bliptext.com/articles/integer).\nThe Takagi–Landsberg curve is a slight generalization, given by\nfor a parameter w; thus the blancmange curve is the case w=1/2.\nThe value H=-\\log_2 w is known as the [Hurst parameter](https://bliptext.com/articles/hurst-parameter).\nThe function can be extended to all of the real line: applying the definition given above shows that the function repeats on each unit interval.\n\n## Functional equation definition\nThe periodic version of the Takagi curve can also be defined as the unique bounded solution to the functional equation\nIndeed, the blancmange function T_w is certainly bounded, and solves the functional equation, since\nConversely, if T:\\R\\to\\R is a bounded solution of the functional equation, iterating the equality one has for any N\nwhence T=T_w.\nIncidentally, the above functional equations possesses infinitely many continuous, non-bounded solutions, e.g.\n\n## Graphical construction\nThe 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.\nIn the illustrations below, progressively finer triangle functions (shown in red) are added to the curve at each stage.\n\n# Properties\n\n## Convergence and continuity\nThe infinite sum defining T_w(x) [converges absolutely](https://bliptext.com/articles/absolute-convergence) for all x. Since for all\nif |w|<1.\nThe Takagi curve of parameter w is defined on the unit interval (or \\mathbb{R}) if |w|<1.\nThe Takagi function of parameter w is [continuous](https://bliptext.com/articles/continuous-function).\nThe functions T_{w,n} defined by the partial sums\nare continuous and [converge uniformly](https://bliptext.com/articles/uniform-convergence) toward T_w:\nfor all x when |w| < 1.\nThis bound decreases as n\\to\\infty.\nBy the [uniform limit theorem](https://bliptext.com/articles/uniform-limit-theorem), T_w is continuous if |w| < 1.\n\n## Subadditivity\nSince the absolute value is a [subadditive function](https://bliptext.com/articles/subadditive-function) so is the function, and its dilations s(2^kx); since positive linear combinations and point-wise limits of [subadditive function](https://bliptext.com/articles/subadditive-function)s are subadditive, the Takagi function is subadditive for any value of the parameter w.\n\n## The special case of the parabola\nFor w=1/4, one obtains the [parabola](https://bliptext.com/articles/parabola): the construction of the [parabola](https://bliptext.com/articles/parabola) by midpoint subdivision was described by [Archimedes](https://bliptext.com/articles/archimedes).\n\n## Differentiability\nFor values of the parameter 0< w < 1/2, the Takagi function T_w is differentiable in the classical sense at any x\\in\\R which is not a [dyadic rational](https://bliptext.com/articles/dyadic-rational).\nBy derivation under the sign of series, for any non dyadic rational x\\in\\R, one finds\nwhere is the sequence of binary digits in the [base 2](https://bliptext.com/articles/base-2) expansion of x:\nEquivalently, the bits in the binary expansion can be understood as a sequence of [square waves](https://bliptext.com/articles/square-wave), the [Haar wavelets](https://bliptext.com/articles/haar-wavelet), scaled to width 2^{-n}.\nThis follows, since the derivative of the triangle wave is just the square wave:\nand so\nFor the parameter 0< w < 1/2, the function T_w is [Lipschitz](https://bliptext.com/articles/lipschitz-continuity) of constant 1/(1-2w).\nIn particular for the special value w=1/4 one finds, for any non dyadic rational x\\in[0,1], according with the mentioned\nFor w=1/2 the blancmange function T_w it is of [bounded variation](https://bliptext.com/articles/bounded-variation) on no non-empty open set; it is not even locally Lipschitz, but it is quasi-Lipschitz, indeed, it admits the function as a [modulus of continuity](https://bliptext.com/articles/modulus-of-continuity).\n\n## Fourier series expansion\nThe Takagi–Landsberg function admits an absolutely convergent Fourier series expansion:\nwith and, for m\\ge 1\nwhere 2^{\\nu(m)} is the maximum power of 2 that divides m. Indeed, the above [triangle wave](https://bliptext.com/articles/triangle-wave) s(x) has an absolutely convergent Fourier series expansion\nBy absolute convergence, one can reorder the corresponding double series for T_w(x):\nputting m=2^n(2k+1) yields the above Fourier series for T_w(x).\n\n## Self similarity\nThe [recursive definition](https://bliptext.com/articles/) allows the [monoid](https://bliptext.com/articles/monoid) of self-symmetries of the curve to be given.\nThis monoid is given by two generators, g and r, which [act](https://bliptext.com/articles/group-action-mathematics) on the curve (restricted to the unit interval) as\nand\nA general element of the monoid then has the form for some integers This [acts](https://bliptext.com/articles/group-action-mathematics) on the curve as a [linear function](https://bliptext.com/articles/linear-function): for some constants a, b and c.\nBecause the action is linear, it can be described in terms of a [vector space](https://bliptext.com/articles/vector-space), with the [vector space](https://bliptext.com/articles/vector-space) basis:\nIn this [representation](https://bliptext.com/articles/group-representation), the action of g and r are given by\nand\nThat is, the action of a general element \\gamma maps the blancmange curve on the unit interval [0,1] to a sub-interval for some integers m, n, p.\nThe mapping is given exactly by where the values of a, b and c can be obtained directly by multiplying out the above matrices.\nThat is:\nNote that is immediate.\nThe monoid generated by g and r is sometimes called the **dyadic monoid**; it is a sub-monoid of the [modular group](https://bliptext.com/articles/modular-group).\nWhen discussing the modular group, the more common notation for g and r is T and S, but that notation conflicts with the symbols used here.\nThe above three-dimensional representation is just one of many representations it can have; it shows that the blancmange curve is one possible realization of the action.\nThat is, there are representations for any dimension, not just 3; some of these give the [de Rham curves](https://bliptext.com/articles/de-rham-curve).\n\n# Integrating the Blancmange curve\nGiven that the [integral](https://bliptext.com/articles/integral) of from 0 to 1 is 1/2, the identity allows the [integral](https://bliptext.com/articles/integral) over any interval to be computed by the following relation.\nThe computation is recursive with computing time on the order of log of the accuracy required.\nDefining\none has that\nThe [definite integral](https://bliptext.com/articles/definite-integral) is given by:\nA more general expression can be obtained by defining\nwhich, combined with the series representation, gives\nNote that\nThis integral is also self-similar on the unit interval, under an action of the dyadic monoid described in the section [Self similarity](https://bliptext.com/articles/).\nHere, the representation is 4-dimensional, having the basis.\nThe action of g on the unit interval is the [commuting diagram](https://bliptext.com/articles/commuting-diagram)\nFrom this, one can then immediately read off the [generators](https://bliptext.com/articles/generating-set-of-a-group) of the four-dimensional representation:\nand\nRepeated integrals transform under a 5,6,... dimensional representation.\n\n# Relation to simplicial complexes\nLet\nDefine the Kruskal–Katona function\nThe [Kruskal–Katona theorem](https://bliptext.com/articles/kruskal-katona-theorem) states that this is the minimum number of (t − 1)-simplexes that are faces of a set of N t-simplexes.\nAs t and N approach infinity,\n(suitably normalized) approaches the blancmange curve.","revisionCount":0},"hasMore":false,"nextCursor":null}

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