Monthly Best Of on 09/28/2023




Artistic view of sixteen pseudo-quaternionic Julia sets ('MandelBulb' like: 'JuliaBulb's)-tridimensional cross-sections-

Jean-François COLONNA
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www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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(CMAP28 WWW site: this page was created on 09/28/2023 and last updated on 11/14/2023 17:50:44 -CET-)




Contents of this page:


1-The 128 most referenced Pictures (*):

(*): Undisplayed pictures -if any- do not exist.



A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the Mandelbrot set border -iteration 5-
1-144 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the von Koch Curve -iteration 2-
2-138 reference(s)		 An elementary monodimensional binary cellular automaton -90- with 49 white starting points -on the bottom line-
3-135 reference(s)		 A Tridimensional Hilbert-like Curve defined with {X2(...),Y2(...),Z2(...)} -iteration 2-
4-134 reference(s)
A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 4-
5-127 reference(s)		 Artistic view of the prime numbers
6-126 reference(s)		 A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 2-
7-125 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the Mandelbrot set border -iteration 1-
8-125 reference(s)
A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the von Koch Curve -iteration 5-
9-123 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 1-
10-121 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the von Koch Curve -iteration 4-
11-121 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the von Koch Curve -iteration 1-
12-121 reference(s)
A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 5-
13-120 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the von Koch Curve -iteration 3-
14-120 reference(s)		 Journey on the Complex Plane by means of a Bidimensional Hilbert Curve -iteration 4- with display of Julia sets
15-119 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the Mandelbrot set border -iteration 4-
16-117 reference(s)
The random walk of photons escaping the Sun
17-116 reference(s)		 A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 3-
18-115 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 5-
19-113 reference(s)		 A Tridimensional Hilbert-like Curve defined with {X1(...),Y1(...),Z1(...)} -iteration 1-
20-113 reference(s)
A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the Mandelbrot set border -iteration 3-
21-113 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the Mandelbrot set border -iteration 2-
22-110 reference(s)		 A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 4-
23-109 reference(s)		 Jean-François COLONNA (on 11/17/1994)with its fractal mountains
24-109 reference(s)
A Tridimensional Hilbert-like Curve defined with {X4(...),Y4(...),Z4(...)} -iteration 4-
25-105 reference(s)		 Journey on the Complex Plane by means of a Bidimensional Hilbert Curve -iteration 1- with display of Julia sets
26-105 reference(s)		 A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 1-
27-104 reference(s)		 Tridimensional representation of a quadridimensional Calabi-Yau manifold
28-103 reference(s)
A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 2-
29-102 reference(s)		 A Tridimensional Hilbert-like Curve defined with {X1(...),Y1(...),Z1(...)} -iteration 1-
30-100 reference(s)		 Untitled 0618
31-99 reference(s)		 A Tridimensional Hilbert-like Curve defined with {X2(...),Y2(...),Z2(...)} -iteration 2-
32-99 reference(s)
A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 3-
33-98 reference(s)		 Journey on the Complex Plane by means of a Bidimensional Hilbert Curve -iteration 3- with display of Julia sets
34-98 reference(s)		 The level-1 cluster made of 9 'Spectre' tiles with display of all the key-points making quadrilaterals (8 small and a big one)
35-97 reference(s)		 Tridimensional display of the 'EinStein' aperiodic 'Spectre' tiling
36-97 reference(s)
A Tridimensional Hilbert-like Curve defined with {X3(...),Y3(...),Z3(...)} -iteration 3-
37-96 reference(s)		 Journey on the Complex Plane by means of a Bidimensional Hilbert Curve -iteration 2- with display of Julia sets
38-96 reference(s)		 A close-up of the 'EinStein' aperiodic 'Spectre' tiling
39-94 reference(s)		 A Tridimensional Hilbert-like Curve defined with {X3(...),Y3(...),Z3(...)} -iteration 3-
40-93 reference(s)
Autostereogram of a quaternionic Julia set -tridimensional cross-section-
41-91 reference(s)		 A Tridimensional Hilbert-like Curve defined with {X4(...),Y4(...),Z4(...)} -iteration 4-
42-91 reference(s)		 A 1972 Télémécanique T1600 computer with a 32 KB central memory and two 512 KB disk drives
43-87 reference(s)		 A random tiling of a square domain using dominoes (1x2 rectangles)-line after line- with display of clusters of vertical rectangles using the 4-connexity
44-85 reference(s)
A random tiling of a square domain using dominoes (1x2 rectangles)-line after line- with display of clusters of horizontal and vertical rectangles using the 4-connexity
45-84 reference(s)		 The 'Mystic' made of two 'EinStein' aperiodic 'Spectre' tiles
46-80 reference(s)		 A Ball described by means of a Tridimensional Hilbert Curve -iteration 4-
47-80 reference(s)		 Tridimensional display of a linear superposition of 6 eigenstates of the Hydrogen atom (tridimensional computation)
48-78 reference(s)
The Lorenz attractor
49-77 reference(s)		 Tridimensional representation of a hexadimensional Calabi-Yau manifold
50-76 reference(s)		 Along the border of the Mandelbrot set
51-75 reference(s)		 The initial conditions of a monodimensional 'quasi-continuous' cellular automata
52-75 reference(s)
Tridimensional representation of a quadridimensional Calabi-Yau manifold described by means of 5x5 Bidimensional Hilbert Curves -iteration 5-
53-74 reference(s)		 Tridimensional representation of a fractal quadridimensional Calabi-Yau manifold
54-74 reference(s)		 An elementary monodimensional binary cellular automaton -106- with random white starting points -on the bottom line-
55-74 reference(s)		 Artistic view of the Cosmic Web (nodes, galaxy clusters, filaments,... including 1.083.984 galaxies)obtained by means of a non deterministic fractal process
56-73 reference(s)
Three successive elementary monodimensional binary cellular automata -106,90,86- with random yellow starting points -on the bottom line-
57-73 reference(s)		 About the length of the von Koch curve
58-72 reference(s)		 Tridimensional representation of quadridimensional Calabi-Yau manifolds -Calabi-Yau manifolds attached to every point of a fractal tridimensional space-
59-72 reference(s)		 Two alternative monodimensional 'quasi-continuous' cellular automata
60-72 reference(s)
The generalized Ulam spiral
61-71 reference(s)		 The alternative use of two elementary monodimensional binary cellular automata -168,165- with periodical white starting points -on the bottom line-
62-71 reference(s)		 An elementary monodimensional binary cellular automaton -90- with 1 white starting point -bottom middle-
63-71 reference(s)		 Iterations in the complex plane: the computation of the Mandelbrot set
64-69 reference(s)
Tridimensional localization of a point P its distances to the four vertices of a tetrahedron ABCD being known
65-69 reference(s)		 Untitled 0616
66-69 reference(s)		 The alternative use of two elementary monodimensional binary cellular automata -168,165- with random white starting points -on the bottom line-
67-69 reference(s)		 Quark and gluon structure of a nucleon
68-68 reference(s)
A tridimensional fractal manifold defined by means of three tridimensional fields
69-67 reference(s)		 Mountains and fog
70-67 reference(s)		 Tridimensional representation of quadridimensional Calabi-Yau manifolds -Calabi-Yau manifolds attached to every point of our familiar tridimensional space?-
71-67 reference(s)		 A Ball described by means of a Tridimensional Hilbert-like Curve -iteration 4-
72-66 reference(s)
Tridimensional display of the Riemann Zeta function inside [-10.0,+20.0]x[-15.0,+15.0] (bird's-eye view)
73-65 reference(s)		 Quantum vacuum fluctuations
74-65 reference(s)		 Some elementary symbols used to built labyrinths -with a big black ghost structure at pi/4-
75-65 reference(s)		 Tridimensional integration of the anisotropic gaussian Fourier filtering of a random field with a 2.pi rotation of the kernel
76-65 reference(s)
Tridimensional representation of quadridimensional Calabi-Yau manifolds -Calabi-Yau manifolds attached to every point of a fractal tridimensional space-
77-65 reference(s)		 An aperiodic Penrose tiling of the plane -a Tribute to Piet Mondrian and Roger Penrose-
78-64 reference(s)		 Botticelli anomaly on the Moon
79-64 reference(s)		 Paradoxal Monument Valley at sunset, 'World of Tiers' -a Tribute to Philip José Farmer-
80-64 reference(s)
The Piet Mondrian quadridimensional Calabi-Yau manifold -2D, 3D or 4D?-
81-64 reference(s)		 A monodimensional 'quasi-continuous' cellular automaton
82-64 reference(s)		 Untitled 0614
83-63 reference(s)		 The numerical irreversibility of the bidimensional billiard
84-63 reference(s)
A medium with percolation -top to bottom- using the 8-connexity
85-63 reference(s)		 Close-up on a foggy pseudo-quaternionic Mandelbrot set (a 'MandelBulb')-tridimensional cross-section-
86-63 reference(s)		 Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb')-tridimensional cross-section-
87-63 reference(s)		 Rotation about the X axis of the Lorenz attractor (1000 iterations), computed simultaneously on two different computers
88-63 reference(s)
Untitled 0617
89-63 reference(s)		 The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 5-
90-63 reference(s)		 Tridimensional representation of a quadridimensional Calabi-Yau manifold described by means of 5x5 Bidimensional Hilbert Curves -iteration 5-
91-63 reference(s)		 Triple impossible staircase and a paradoxal structure
92-63 reference(s)
A monodimensional 'quasi-continuous' cellular automaton
93-63 reference(s)		 Coalescence of 40832 particles -black holes?- inside a tridimensional parallelepipedic billiard
94-62 reference(s)		 Empty
95-62 reference(s)		 The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 5-
96-62 reference(s)
A Ball described by means of an hypercube -iteration 4-
97-62 reference(s)		 A torus described by means of a Bidimensional Hilbert Curve -iteration 5-
98-62 reference(s)		 Bidimensional Hilbert Curve -iteration 5-
99-62 reference(s)		 A random tiling of a square domain using dominoes (1x2 rectangles)-line after line-
100-62 reference(s)
The initial conditions of a monodimensional 'quasi-continuous' cellular automata
101-62 reference(s)		 About the fractal dimension with the first two iterations -red and green respectively- of the construction of the von Koch curve
102-61 reference(s)		 Bidimensional display of the rounding-off errors when computing the Verhulst dynamics
103-61 reference(s)		 The Universe at the heart of a Calabi-Yau manifold
104-61 reference(s)
Dissonance chaude/Dissonance froide -a Tribute to Paul Sérusier-
105-61 reference(s)		 N-body problem integration (N=4: one star, one heavy planet and one light planet with a satellite)computed with 2 different optimization options on the same computer (sensitivity to rounding-off errors)
106-61 reference(s)		 Tridimensional visualization of the Mandelbrot set with mapping of the arguments
107-61 reference(s)		 The Lorenz attractor -sensitivity to integration methods used (Red=Euler, Green=Runge-Kutta/2nd order, Blue=Runge-Kutta/4th order)-
108-61 reference(s)
Untitled 0612
109-61 reference(s)		 The Ptolemaic system with equant -grey circle above the Earth- and with a small light grey circle -the epicycle- whose center describes a larger dark grey circle -the deferend-
110-61 reference(s)		 An aperiodic Penrose tiling of the plane -a Tribute to Piet Mondrian and Roger Penrose-
111-60 reference(s)		 Autostereogram with an hidden volcano
112-60 reference(s)
True colors autostereogram of a quaternionic Julia set -tridimensional cross-section-
113-60 reference(s)		 A foggy pseudo-quaternionic Mandelbrot set (a 'MandelBulb')-tridimensional cross-section-
114-60 reference(s)		 A pseudo-octonionic Mandelbrot set (a 'MandelBulb')-tridimensional cross-section-
115-60 reference(s)		 Bidimensional visualization of the Verhulst dynamics -(grey,orange,red)display negative Lyapunov exponents, (yellow,green,blue) display positive Lyapunov exponents-
116-60 reference(s)
Rotation about the X axis of the Lorenz attractor (5000 iterations), computed simultaneously on two different computers
117-60 reference(s)		 The Piet Mondrian Hypercube -2D, 3D or 4D?-
118-60 reference(s)		 A Ball described by means of an 'open' 3-foil torus knot -iteration 4-
119-60 reference(s)		 Triple impossible staircase painting a paradoxal structure
120-60 reference(s)
Rotation about the Y (vertical)axis of a tridimensional representation of a quadridimensional Calabi-Yau manifold that can also be viewed as a set of 4x3 stereograms
121-60 reference(s)		 Universe or Multiverse -The fractal Universe-?
122-59 reference(s)		 An aperiodic Penrose tiling of the Golden Decagon -a Tribute to Piet Mondrian and Roger Penrose-
123-59 reference(s)		 A tridimensionally distorded -by means of a fractal bidimensional height field- aperiodic Penrose tiling of the Golden Decagon
124-59 reference(s)
Fractal diffusion front in a bidimensional medium obtained by means of identical interacting particles
125-59 reference(s)		 Quaternionic butterfly with extended arithmetics
126-59 reference(s)		 The Bonan-Jeener double bottle described by means of a Bidimensional Hilbert-like Curve -iteration 5-
127-59 reference(s)		 The Solar System with a dark blue virtual planet -virtual planet point of view-
128-59 reference(s)




2-The 128 most referenced Pages (*):

(*): Unclickable pages -if any- do not exist.



1-AVirtualSpaceTimeTravelMachine.Ang
1350 reference(s)		 2-An2000.01.Fra
147 reference(s)
3-demo_14
140 reference(s)		 4-Vcatalogue.11
135 reference(s)
5-LeNoeudInfini.01.Fra
122 reference(s)		 6-Informations_AboutPicturesAnimationsAndFiles.01.Ang
120 reference(s)
7-ChatGPT.01.Fra
112 reference(s)		 8-Galerie_NumberTheory.FV
109 reference(s)
9-help.
107 reference(s)		 10-copyright.01.
98 reference(s)
11-LeNoeudInfini.01.Ang
98 reference(s)		 12-LesCourbesRemplissantes.02.Fra
97 reference(s)
13-mail.01.vv
95 reference(s)		 14-Web_Mail.01.vv
94 reference(s)
15-Fractal.01
94 reference(s)		 16-catalogue.11
86 reference(s)
17-Galerie_GeneralitiesVisualization.FV
84 reference(s)		 18-FloatingPointNumbers.01.Ang
84 reference(s)
19-Stereogrammes_AutoStereogrammes.01
83 reference(s)		 20-NDimensionalDeterministicFractalSets.01.Fra
82 reference(s)
21-Galerie_DeterministicFractalGeometry.FV
82 reference(s)		 22-AProposSite.01.Fra
77 reference(s)
23-AVirtualSpaceTimeTravelMachine.Fra
76 reference(s)		 24-xiirv_____.nota.t.
75 reference(s)
25-An2000.01.Ang
75 reference(s)		 26-Galerie_ImagesDesMathematiques.FV
74 reference(s)
27-AnimFractal.01.
74 reference(s)		 28-De_L_EnseignementAssisteParOrdinateur_a_L_ExperimentationVirtuelle.01.Fra
73 reference(s)
29-LesCourbesRemplissantes.02.Ang
71 reference(s)		 30-Informations_GoodNewsAndBadNews.01.Fra
71 reference(s)
31-GoldenTriangle.01.Fra
71 reference(s)		 32-FloatingPointNumbers.01.Fra
70 reference(s)
33-EntrelacsIntertwinings.01.Fra
70 reference(s)		 34-SourceExemple_____xrq_____hydrogene.61.K.
69 reference(s)
35-LOG_xiMc.11
69 reference(s)		 36-Galerie_ArtAndScience.FV
69 reference(s)
37-Polynomials_RationalNumbers.01.Fra
67 reference(s)		 38-Galerie_TextureSynthesis.FV
67 reference(s)
39-SurfaceProjector.01.Fra
66 reference(s)		 40-MathematiquesPhysiqueFractales.22
66 reference(s)
41-Galerie_NonDeterministicFractalGeometryNaturalPhenomenonSynthesis.FV
66 reference(s)		 42-DieuScience.01.Ang
66 reference(s)
43-DecimalesDePi.01.Fra
66 reference(s)		 44-LesCourbesRemplissantes.01.Ang
64 reference(s)
45-ImagesDuVirtuel.01.Fra
64 reference(s)		 46-Galerie_ArtisticCreation.FV
64 reference(s)
47-AVirtualSpaceTimeTravelMachine.Ang.FV
64 reference(s)		 48-MonodimensionalCellularAutomata.01.Fra
63 reference(s)
49-Exposition_EcolePolytechnique_FeteDeLaScience_201910
63 reference(s)		 50-ArithmetiqueEtOrdinateur.01.vv.Fra
63 reference(s)
51-MathematiquesPhysiqueFractales.12
62 reference(s)		 52-Galerie_SensitivityToRoundingOffErrors.FV
62 reference(s)
53-Fractal.11
62 reference(s)		 54-FaireDeLaRecherche.01.vv
62 reference(s)
55-AProposSite.01.Ang
61 reference(s)		 56-RealNumbers.01.Fra
60 reference(s)
57-MouvementsRelatifs_et_ObservationsAstronomiques.01.Fra
60 reference(s)		 58-ImagesDuVirtuel.01.Fra.FV
60 reference(s)
59-GoldenTriangle.01.Ang
60 reference(s)		 60-Galerie_ParticleSystems.FV
60 reference(s)
61-Ariane501.01.Fra
60 reference(s)		 62-AQuoiServentLesMathematiques.02
60 reference(s)
63-MathematiquesPhysiqueFractales.02
59 reference(s)		 64-Galerie_ImagesDidactiques.FV
59 reference(s)
65-UlamSpiral.01.Fra
58 reference(s)		 66-SurfaceProjector.01.Ang
58 reference(s)
67-MathematiquesPhysiqueFractales.02.23
58 reference(s)		 68-Galerie_Astrophysics.FV
58 reference(s)
69-Labyrinthes.01.Ang
57 reference(s)		 70-IllusionConnaissance.02
57 reference(s)
71-Fractal.21
57 reference(s)		 72-ExpV_VirE.11
57 reference(s)
73-DieuScience.01.Fra
57 reference(s)		 74-Commentaires_VarieteHexaDecaDimensionnelleCalabiYau.01
57 reference(s)
75-Commentaires_ProblemeDesNCorps.01
57 reference(s)		 76-LesCourbesRemplissantes.01.Fra
56 reference(s)
77-EntrelacsIntertwinings.01.Ang
56 reference(s)		 78-DecimalesPi_1_100.vv
56 reference(s)
79-An2000.04.Fra
56 reference(s)		 80-RemarquesCerveau.01
55 reference(s)
81-PerteDeLAssociativite.01
55 reference(s)		 82-Notations_AppartientA.01
55 reference(s)
83-Multivers.01.Fra
55 reference(s)		 84-infinity.01.vv
54 reference(s)
85-NatureDesMathematiques.01.vv.Fra
54 reference(s)		 86-GenieLogiciel.01.Fra
54 reference(s)
87-Galerie_CelestialMechanics.FV
54 reference(s)		 88-Exposition_MairieCinquiemeArrondissementParis_202001_202002
54 reference(s)
89-ExpV_VirE.Fra
54 reference(s)		 90-AQuoiServentLesMathematiques.01
54 reference(s)
91-Pcatalogue.11
53 reference(s)		 92-OrdinateurMathematiquesArt.01.Fra
53 reference(s)
93-NombresPremiers_10000.vv
53 reference(s)		 94-GenieLogiciel_VisualisationScientifique.01.vv
53 reference(s)
95-Galerie_QuantumMechanics.FV
53 reference(s)		 96-FaireDesMathematiques.01.Fra
53 reference(s)
97-Exposition_EcolePolytechnique_FeteDeLaScience_201810
53 reference(s)		 98-Exposition_EcolePolytechnique_AnneeMondialeDeLaPhysique_2005
53 reference(s)
99-present.01.
52 reference(s)		 100-Informations_AboutPicturesAnimationsAndFiles.01.Fra
52 reference(s)
101-Galerie_AnthropomorphicPatterns.FV
52 reference(s)		 102-PerteDeLaDistributivite.01.DIS2
51 reference(s)
103-Notations_QuelQueSoit.01
51 reference(s)		 104-NDimensionalDeterministicFractalSets.01.Ang
51 reference(s)
105-LeParadoxeDeFermi.01.Fra
51 reference(s)		 106-DuReelAuVirtuel.01.Fra
51 reference(s)
107-Commentaires_EnsembleMandelbrotQuaternion.01
51 reference(s)		 108-Commentaires_DefinitionComplexes.01
51 reference(s)
109-ArtScience.01
51 reference(s)		 110-An2000.05.Fra
51 reference(s)
111-AVirtualSpaceTimeTravelMachine.Ang.m4
51 reference(s)		 112-MathematiquesPhysiqueFractales.02.14
50 reference(s)
113-IllusionConnaissance.01
50 reference(s)		 114-Galerie_SelfPortraits.FV
50 reference(s)
115-Commentaires_DefinitionQuaternions.01
50 reference(s)		 116-ArtScience.11.Ang.DIAPO.0001
50 reference(s)
117-T_PerteDeLAssociativite.01_ProblemesNombresFlottants.03.0001
49 reference(s)		 118-RasoirDOccamEtMathematiques.01.vv.Fra
49 reference(s)
119-MathematiquesPhysiqueFractales.02.28
49 reference(s)		 120-MathematiquesPhysiqueFractales.02.22
49 reference(s)
121-MathematiquesPhysiqueFractales.02.13
49 reference(s)		 122-Labyrinthes.01.Fra
49 reference(s)
123-HommageBenoitMandelbrot.21
49 reference(s)		 124-Galerie_DeterministicChaos.FV
49 reference(s)
125-Exposition_EcolePolytechnique_FeteDeLaScience_201510
49 reference(s)		 126-Commentaires_VarieteQuadriDimensionnelleCalabiYau.01
49 reference(s)
127-Commentaires_EnsembleMandelbrotOctonion.01
49 reference(s)		 128-Commentaires_DefinitionPseudoQuaternions.01
49 reference(s)

And now, enjoy visiting A Virtual Space-Time Travel Machine.




Copyright © Jean-François COLONNA, 2023-2023.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2023-2023.