• NameSurfer
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• SURFER
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• SURFER-2008
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• QI Touchviewer
  • START (WEB 1)
  • START (WEB 2)
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• 3D-XplorMath
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• • Espacio Euclideano R3: Sistema de coordenadas tridimensional: x, y, z

• • Podemos definir una superficie como una ecuación: F(x,y,z)=0
    • por ejemplo x^2 + y^2 + z^2 -4 = 0
  • SURFER muestra las superficies en una zona esférica

• • Lo que sobresale de la esfera se «recorta»

• Probar:
  • x^2+y^2+z^2+1500*(x^2+y^2)*(x^2+z^2)*(y^2+z^2)-1
• Cambiar un + por –
  • x^2+y^2+z^2+1500*(x^2+y^2)*(x^2-z^2)*(y^2+z^2)-1

• Nombres o palabras que producen curvas bonitas
  • Alejandro Alicante Anacleto Antonio Armando Australopitecus Cervantes Dymitri Ester Etelvina Feliciano Felipe Ildefonso Lucía
    Maricarmen Naranja Paco Patata Pedro Pepe Peter Piojo Sebastián Tabarca Valencia Xavier
    • Algunas imágenes

• • PLANOS:
    • Un plano x=0
      • x
    • Dos planos x=0 e y=0
      • x*y
    • Tres planos
      • x*y*z
  • CILINDRO:
    • Cilindro en sentido de la z
      • (x^2+y^2-1)
    • Dos cilindros cruzados
      • (x^2+y^2-1)*(x^2+z^2-1)
    • Con un exponente impar
      • (x^2+y^2-1)*(x^2+y^3-4)
      • cambiar la primera x por z
      • cambiar la primera y por z
    • Tres cilindros: x, y, z
      • (x^2+y^2-1)*(x^2+z^2-1)*(y^2+z^2-1)
    • Tres caños cuadrados
      • (x^6+y^6-1)*(x^6+z^6-1)*(z^6+y^6-1)
  • PARÁBOLA:
    • x + y + z^2
  • ESFERA:
    • Esfera

Cuando el centro es el origen de coordenadas la ecuación que deben satisfacer los puntos P(x,y,z) para pertenecer a
la esfera es: x^2 + y^2 + z^2 = r^2

• • • • por ejemplo
        • x^2 + y^2 + z^2 -4
        • ¿Cómo dibujarías un planeta con anillo como Saturmo?
      • cambiar un más por menos
        • x^2+y^2-z^2-0.5
      • agregar parámetro +a
        • x^2+y^2-z^2+a-0.5
      • cambiar +a por *a
        • x^2+y^2-z^2*a-0.5
      • quitar el término independiente
        • x^2+y^2-z^2
      • cambiar el exponente de z a 3
        • x^2+y^2-z^3
      • platillo de batería
        • x^2+y^2-z^9
      • Dos esferas
        • (x^2+y^2+z^2-1)*((x-3*a)^2+y^2+z^2-1)
    • Gragea
      • x^2 + y^2 + 9*z^2 -4

• • • Píldora
      • 8*x^2 + 8*y^2 + z^4 -3

• • Hamburguesa
    • x^4+y^2+z^2-19
  • Cilindro (con halo)
    • x^2 + y^2 + z^28 -4
    • Hacer zoon hasta que intersecte la esfera externa
  • Sombrero Andalúz
    • x^2 + y^2 + z^23 -3
  • Capelina (expionente impar)
    • x^2 + y^2 + z^3 -3
  • Capelina cubista
    • x^23 + y^23 + z^23 -3
  • Cubo
    • x^6+y^6+z^6-1
  • Combinando superficies (multiplicar)
    • z
    • (x^2+y^2-1)
    • (x^2+y^2+z^3-4)
    • (x^2+y^2+z^3-4)*(x^2+y^2-1)
    • (x^2+y^2-1)*z*(x^2+y^2+z^3-4)*(x^2+y^2-z^3-4)*x
    • x*y*z*(x^6+y^6-1)*(y^6+z^6-1)*(x^6+z^6-1)*(x^2+y^2+z^3-4)*(x^2+y^2-z^3-4)
    • x*y*z*(x^6+y^6-1)*(y^6+z^6-1)*(x^6+z^6-1)*(x^2+y^2+z^3-4)*(x^2+y^2-z^3-4)*(x^2+z^2+y^3-4)*(x^2+z^2-y^3-4)*(z^2+y^2+x^3-4)*(z^2+y^2-x^3-4)
    • dos esferas
      • (x^2+y^2+z^2-1)*((x-3*a)^2+y^2+z^2-1)
  • Pincho Moruno
    • (x^8+y^8+(z+4)^8-8)*(x^2+y^2+z^2-4)*(x^8+y^8+(z-4)^8-8)*(x^2+y^2-0.01)

• • Tick (Peonza)
    • x^2+y^2-z^3*(1-z)
  • Alex
    • x*(x-1)*(x+1)*y+z*(x-1)*(x+1)
  • Limón
    • 1.2*x^2+1.2*z^2-5*(y+0.5)^3*(0.5-y)^3

• • Corazón
    • x^2+z^3*a^4
      • Hay que buscarle el tamaño y ángulo adecuado
    • (x^2+9/4*y^2+z^2-1)^3-x^2*z^3-9/80*y^2*z^3
  • Donut
    • (x^2+y^2+z^2+a^2-b^2)^2-4*b^2*(x^2+y^2)

• • Chmutov Octics
    • -2*a/125 + x^8 + y^8 + z^8 – 2*x^6 – 2*y^6 – 2*z^6 + 1.25*x^4 + 1.25*y^4 + 1.25*z^4 – 0.25*x^2 – 0.25*y^2 – 0.25*z^2 + 0.03125
    • Chmutov’s octic equation written by SV Chmutov in the early 80s. At the time, it constituted the world record for μ(d) most
      d. Beginning numbers have been replaced by some of the Fibonacci numbers sequence. Full description of the process is available
      on the following page http://www.hermay.org/jconstant/surfer/resource.html
    • 0 * z^3 – 5 * x^8 * z^13 – 24 * y^2 * z^6 + 36 * z^8 + 24 * x^4 * z^3 – 168 * x^2 * y^2 * z^3 + 24 * y^4 * z^3 – 72 * x^2
      * z^5 – 72 * y^2 * z^5 + 54 * z^7 – 8 * x^6 – 24 * x^4 * y^2 – 24 * x^2 * y^4 – 8 * y^6 + 36 * x^4 * z^2 – 252 * x^2 * y^2
      * z^2 + 36 * y^4 * z^2 – 54 * x^2 * z^4 – 108 * y^2 * z^4 + 27 * z^6 – 108 * x^2 * y^2 * z + 54 * y^4 * z – 54 * y^2 * z^3
      + 27 * y^4
    • (a * ( – 55 * 34)) / 21 + x^13 + y^8 + z^5 – 3 * 2 * x^1 – 1 * 2 * y^6 – 1 * 2 * z^6 + 1.25 * x^4 + 1.25 * y^4 + 1.25 * z^4
      – 1 * 0.25 * x^2 – 1 * 0.25 * y^2 – 1 * 0.25 * z^2 + 0.03125
    • 01. Quaste «the ABC of an equation of Cartesian nature. Monomials of degree 1 are x, y, z. Monomials of degree 2 are x2 ,
      xy, y2, yz, z2, etc»
      • 8 * z^9 – 24 * x^2 * z^6 – 24 * y^2 * z^6 + 36 * z^8 + 24 * x^4 * z^3 – 168 * x^2 * y^2 * z^3 + 24 * y^4 * z^3 – 72 * x^2
        * z^5 – 72 * y^2 * z^5 + 54 * z^7 – 8 * x^6 – 24 * x^4 * y^2 – 24 * x^2 * y^4 – 8 * y^6 + 36 * x^4 * z^2 – 252 * x^2 * y^2
        * z^2 + 36 * y^4 * z^2 – 54 * x^2 * z^4 – 108 * y^2 * z^4 + 27 * z^6 108 * x^2 * y^2 * z + 54 * y^4 * z – 54 * y^2 * z^3 +
        27 * y^4
    • 02. Quaste #2
      • 8 * z^9 – 24 * x^2 * z^6 – 24 * y^2 * z^6 + 36 * z^8 + 24 * x^4 * z^3 – 168 * x^2 * y^2 * z^3 + 24 * y^4 * z^3 – 72 * x^2
        * z^5 – 72 * y^2 * z^5 + 54 * z^7 – 8 * x^6 – 24 * x^4 * y^2 – 24 * x^2 * y^4 – 8 * y^6 + 36 * x^4 * z^2 – 252 * x^2 * y^2
        * z^2 + 36 * y^4 * z^2 – 54 * x^2 * z^4 – 108 * y^2 * z^4 + 27 * z^6 – 108 * x^2 * y^2 * z + 54 * y^4 * z – 54 * y^2 * z^3
        + 27 * y^4
    • 03. Quaste #3
      • 0 * z^3 – 5 * x^8 * z^13 – 24 * y^2 * z^6 + 36 * z^8 + 24 * x^4 * z^3 – 168 * x^2 * y^2 * z^3 + 24 * y^4 * z^3 – 72 * x^2
        * z^5 – 72 * y^2 * z^5 + 54 * z^7 – 8 * x^6 – 24 * x^4 * y^2 – 24 * x^2 * y^4 – 8 * y^6 + 36 * x^4 * z^2 – 252 * x^2 * y^2
        * z^2 + 36 * y^4 * z^2 – 54 * x^2 * z^4 – 108 * y^2 * z^4 + 27 * z^6 – 108 * x^2 * y^2 * z + 54 * y^4 * z – 54 * y^2 * z^3
        + 27 * y^4
    • 04. Chmutov Octic. Equation written by SV Chmutov in the early 80s. At the time it constituted the world record for μ(d)
      most d. Beginning numbers have been replaced by some of the Fibonacci numbers sequence.
      • (a * ( – 55 * 34)) / 21 + x^13 + y^8 + z^5 – 3 * 2 * x^1 – 1 * 2 * y^6 – 1 * 2 * z^6 + 1.25 * x^4 + 1.25 * y^4 + 1.25 * z^4
        – 1 * 0.25 * x^2 – 1 * 0.25 * y^2 – 1 * 0.25 * z^2 + 0.03125
    • 05 Chmutov octic #2 on a vis a vis background disk. The cusp on the surface is a singularity. Black holes and the Big Bang
      constitute singularities cosmological model equations. The original equation has been altered with reverse Fibonacci numbers
      sequence.
      • x^5-x^3+y^2+y^1+z^1-z^0
      • (a * ( – 55 * 34)) / 21 + x^13 + y^8 + z^5 – 3 * 2 * x^1 – 1 * 2 * y^6 – 1 * 2 * z^6 + 1.25 * x^4 + 1.25 * y^4 + 1.25 * z^4
        – 1 * 0.25 * x^2 – 1 * 0.25 * y^2 – 1 * 0.25 * z^2 + 0.03125
    • 06 Chmutov Octic #3 . Variation on the Chmutov octic equation that brings up its symmetrical aspect. Included, a parabola.
      • x*y*(x^2-y)
    • 07 RGB. Two spheres. Same object different color. Each as a distinct identity according to the RGB color model .
      • (x^1+y^1+z^2-2)*((x-3*a)^2+y^2+z^2-1)
  • Barth Sextic with 30 Cusps
    • 4 * ((a * (1 + sqrt(5)) / 2)^2 * x^2 – y^2) * ((a * (1 + sqrt(5)) / 2)^2 * y^2 – z^2) * ((a * (1 + sqrt(5)) / 2)^2 * z^2 –
      x^2) – (1 + 2 * (a * (1 + sqrt(5)) / 2)) * (x^2 + y^2 + z^2 – 1)^3
    • http://2048.imaginary.org/images-linked/256.gif
  • Kummer Quartic
    • (x^2 + y^2 + z^2 – (0.5 + 2 * a)^2)^2 – (3.0 * ((0.5 + 2 * a)^2) – 1.0) / (3.0 – ((0.5 + 2 * a)^2)) * (1 – z – sqrt(2) * x)
      * (1 – z + sqrt(2) * x) * (1 + z + sqrt(2) * y) * (1 + z – sqrt(2) * y)
    • http://2048.imaginary.org/images-linked/64.gif
  • Ta-Te-Ti
    • tictactoeb_bridges2015.pdf
    • (x-1)*(x+1)*(y-1)*(y+1)*(x^2+y^2-0.2)
    • (x-a)*(x+a)*(y-b)*(y+b)*(x^2+y^2-0.2)*((x-2)^2+(y+2)^2+z^2-0.2)
  • ?
    • ((b * x^2 + (y – 0.3)^2 + z^2 – 0.5) * (c * x^2 + y^2 + z^2 – 0.5) * (d * x^2 + (y – 0.5)^2 + z^2 – 0.1) – a)^2
  • Alien vs Virus
    • alienvsvirus_bridges2015_0.pdf
    • (x^2+y^2-1)^2+(z^2+(y+3-6*b)^2-1)^2-0.01*a
    • ((b*x^2 + (y – 0.3)^2 + z^2 – 0.5)*(c*x^2 + y^2 + z^2 – 0.5)*(d*x^2 + (y – 0.5)^2 + z^2 – 0.1) – a)^2 + (x*y*z)^2 – 0.000001
    • (6*x^2 – 2*x^4 – y^2*z^2)^2 + (8*z^9 – 24*x^2*z^6 – 24*y^2*z^6 + 36*z^8 + 24*x^4*z^3 – 168*x^2*y^2*z^3 + 24*y^4*z^3 – 72*x^2*z^5
      – 72*y^2*z^5 + 54*z^7 – 8*x^6 – 24*x^4*y^2 – 24*x^2*y^4 – 8*y^6 + 36*x^4*z^2 – 252*x^2*y^2*z^2 + 36*y^4*z^2 – 54*x^2*z^4 –
      108*y^2*z^4 + 27*z^6 – 108*x^2*y^2*z + 54*y^4* z – 54*y^2*z^3 + 27*y^4)^2 – 0.01
  • Togliatti Quintic
    • (1 – sqrt(5 – sqrt(5)) / 2 * z) * (x^2 + y^2 – 1 + (1 + 3 * sqrt(5)) / 4 * z^2)^2 – a * 3.8496061120482923113983843766 * (x
      – z) * (cos(2 * 3.14159265358979 / 5) * x – sin(2 * 3.14159265358979 / 5) * y – z) * (cos(4 * 3.14159265358979 / 5) * x –
      sin(4 * 3.14159265358979 / 5) * y – z) * (cos(6 * 3.14159265358979 / 5) * x – sin(6 * 3.14159265358979 / 5) * y – z) * (cos(8
      * 3.14159265358979 / 5) * x – sin(8 * 3.14159265358979 / 5) * y – z) * 001
    • Togliatti Quintic
      • A classical question in algebraic geometry is how many cone-singularities a surface of a given degree can have. In 1940 Togliatti
        proved that for degree 5 surfaces 31 cone-singularities are possible. It took 40 years until Beauville could prove that this
        is indeed the maximal number. The movie shows a quintic with 31 cone-singularities.
    • http://2048.imaginary.org/images-linked/128.gif
  • Solid Trefoil Knot (Pretzel)
    • Surfer_Trefoil-Stephan-Klaus.pdf
    • http://data.imaginary-exhibition.com/IMAGINARY-Trefoil-Stephan-Klaus.pdf
    • (-8*(x^2 + y^2)^2*(x^2 + y^2 + 1 + z^2 + a^2 – b^2) + 4*a^2*(2*(x^2 + y^2)^2 – (x^3 – 3*x*y^2)*(x^2 + y^2 + 1)) + 8*a^2*(3*x^2*y
      – y^3)*z + 4*a^2*(x^3 – 3*x*y^2)*z^2)^2 – (x^2 + y^2)*(2*(x^2 + y^2)*(x^2 + y^2 + 1 + z^2 + a^2 – b^2)^2 + 8*(x^2 + y^2)^2
      + 4*a^2*(2*(x^3 – 3*x*y^2) – (x^2 + y^2)*(x^2 + y^2 + 1)) – 8*a^2*(3*x^2y – y^3)*z – 4*(x^2 + y^2)*a^2*z^2)^2
  • surfer-images-printing-samples.pdf
    • x^4*y^3
      • (usar Surfer 2008 y aún así va lento)

• • Yin-Yang
    • (y)*(z^3+(x-2)*x*(x+2))*((x-1)^2+y^2+y^2+(z-0.2)^2-a*2)^2*((x+1)^2+y^2+(z+0.2)^2-2*a)^2
  • 7*x*z^4-x^2-y^2+3
  • x^4*y^2+y^4*x^2-x^2*y^2+z^6
  • x^2*y*z+x*y^2+y^3+y^3*z-x^2*z^2
  • ((x*y-z^3-1)^2+(x^2+y^2-1)^3)^2*(z+z*y*2)^4
  • Flor
    • (x^2*y^2-(z-25)^3)*(x^2+y^2-0.1)
  • Tulipán
    • (((z-2)^3-2)^2+(x^2+(y+2)^2-3)^3)*(x^2+(y+2)^2-0.002)
  • (x^2+y^2)^3-4*x^2*y^2*(z^2+1)
  • Trebol
    • ((x^2+y^2)^3-4*x^2*y^2*(z^2+1))*(y^2+z^2 -8)
  • (((z-2)^3-2)^2+(x^2+(y^2)^2-3)-4*x^2*y^2*(z^2+1))*(y^2+z^2 -8)
• Worldrecordsurfaces – Algebraic surfaces with many singularities (PDF)

• opc 1
• opc 2

• 3D-XplorMath-J.jar
• Surfaces of Constant Width

• SURFER
  • Red Blub (Vimeo)
  • YinYan (Vimeo)
  • IMAGINARY – CosmoCaixa Madrid (YouTube)
  • Surfer Video / IMAGINARY 2008 (YouTube)
    • The SURFER program is a unique realtime raytracer of algebraic surfaces with an intuitive user interface. It was developed
      for the traveling Math Art Exhibition IMAGINARY 2008 by the Mathematisches Forschungsinstitut Oberwolfach with support buy
      the Technical University Kaiserslautern. It became very popular – over 10 000 pictures and videos have been created with SURFER
      by over 3000 people in just one year. MATH ART has become MASS ART!
    • Surfer (bis) (vimeo
  • Reise durch das Weltall (Vimeo)
  • Surfer – Imaginary (Vimeo)
    • Surfer – Imaginary (bis)
  • IMAGINARY in the Deutsches Museum, Munich (Vimeo)
  • heart/colibri morph (Vimeo)
  • Fractal Audio (Surfer)
    • Animation film made by Mariano Merchante as part of the lecture «Interactive Art and Science», ITBA, Buenos Aires 2011. It
      was one of the first excercices of the lecture to play with Surfer and create an animation. The music is by Mindthings, the
      song called Fractal.
  • NameSurfer (YouTube)
  • Metamorphose (Vimeo)
• QI-Touchviewer
  • QI-Touchviewer (Vimeo)
• 3D-XplorMath
  • Surfaces of Constant Width (Vimeo)
• OTROS
  • Algebraic Vibration (Vimeo)
  • Katzengold (Vimeo)
  • Videos by Bianca Violet (Vimeo)

• Matemáticas y arte (Wikipedia)
• Guía Surfer 2015 (Uruguay)
• Lo que se puede hacer con Surfer: insecto
• Exposición Imaginary-Surfer

• 2048

• Acceso a estos slides
  • http://alexbia.umh.es/talleres-jpa/arte-y-mates/
    • https://bit.ly/3vEOYnI