"Marching Waves" Sinusoidal toroid. (66a) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Driftwood Wreath" Toroid with sinusoidally modulated radius. (70b) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Sinuous Signal" Toroid with sinusoidally modulated radius. (79) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Balanced Bulb" Lobed torus. (80) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Tidal Twist" Coiled toroid with sinusoidally modulated radius. (78b) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Tidal Twist" Coiled toroid with sinusoidally modulated radius. (78a) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Star of Symmetry" Lobed double toroid. (72c) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Trifold Flow" (39) Toroid with three sets of four strands. #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Clashing Curves" Sinusoidal toroid with three sets of nine strands. (49) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Fivefold Fields" Coiled toroid with five sets of five strands. (15) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Parametric Turbulence" Coiled torus with three sets of ten strands. (76) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Birotary Annuli" Two ring tori with opposite directions of rotation. (46) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Dizzy Donut" Ring torus with two strands. (32) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Four Winds" Coiled torus with four sets of two strands. (31) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Harmonic Crown" Lobed torus with ten strands and sinusoidally modulated radius. Created in Excel. (74a) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Harmonic Crown" Lobed torus with ten strands and sinusoidally modulated radius. (75b) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

"Harmonic Crown" Lobed torus with ten strands and sinusoidally modulated radius. (75a) #CreativeCoding #DigitalArt #MathArt #MathSky #SciArt

(animation 20) #MathArt

(animation 48) #MathArt

(animation 14) #MathArt

(animation 27) #MathArt

(animation 38) #MathArt

n=0,1,2,3,... f(n,r,s,t)=⌊n/(r*s)⌋ mod t Creates a sequence of integers from 0 to t-1 that repeats each digit r*s number of times. In Excel: MOD(INT(n/(r*s)),t)

(animation 19) #MathArt

(animation 18) #MathArt

n=0,1,2,3,... f(n,s,t)=⌊n/t⌋ mod s Creates a repeating sequence of integers from 0 to s-1 where each integer is repeated t number of times. Excel formula: MOD(INT(n/t),s)

n=0,1,2,3,... f(n,t)=⌊n/t⌋ Creates a sequence starting at 0 where each integer is repeated t number of times. In Excel: INT(n/t)

n=0,1,2,3,... f(n,t)=n mod t Creates a repeating sequence of integers from 0 to t-1. In Excel: MOD(n,t)

(animation 13) #MathArt

xz plane view (80 degree x-axis rotation). 4 rotating groups of 4 rotating draw points with no twisting creating 16 circles aligned with the xy plane. Basic graphing of my revised parametric equations for multi-strand torus windings. (animation 33d) #MathArt

n=0,1,2,3,... 2*((n mod t) mod 2)-1 A sequence of -1,1,... that progresses t number of times before starting over. In Excel: 2*MOD(MOD(n,t),2)-1

a=25 (strands/vertices in polygon 1) b=1 (strands/vertices in polygon 2) c=.35 (radius of polygon 1) d=.175 (radius of polygon 2) f=.525 (torus major radius) g=0 (twist rate for polygon 1) h=4 (twist rate for polygon 2) #MathArt

a=3 b=4 c=.2 d=.05 e=.0075 f=.8 g=5 h=30 k=(π/180) n=0,1,2,3,... p=⌊(n/(a*b))⌋ r=(n mod a)*(2*π/a) s=(n mod b)*(2*π/b) t=n*e*k u=⌊(n/a)⌋ v=⌊(n/b)⌋ i=t*g j=t*h x=((.5*f+c*cos(i+u*r))+(.5*f+d*cos(j+v*s)))*cos(t) y=((.5*f+c*cos(i+u*r))+(.5*f+d*cos(j+v*s)))*sin(t) z=c*sin(i+u*r)+d*sin(j+v*s)

a=2 b=4 c=.2 d=.1 e=.0075 f=.8 g=5 h=30 k=(π/180) n=0,1,2,3,... p=⌊(n/(a*b))⌋ r=(p mod a)*(2*π/a) s=(n mod b)*(2*π/b) t=n*e*k u=⌊(n/a)⌋ v=⌊(n/b)⌋ i=n*e*g*k j=n*e*h*k x=((.5*f+c*cos(i+u*r))+(.5*f+d*cos(j+v*s)))*cos(t) y=((.5*f+c*cos(i+u*r))+(.5*f+d*cos(j+v*s)))*sin(t) z=c*sin(i+u*r)+d*sin(j+v*s)

I just want to explore all the possibilities for parametric equations. What else should I try? #MathArt

s=5 (number of lines) q=10 (revolutions increment) d=.0075 (degree step) n=0,1,2,3,... r=.2*sin(4*v)+.4 a=int(n/s) t= n (mod s) * (2*3.14159/s) u=a*d*(3.14159/180) v=n*d*(3.14159/180) b=n*d*q*(3.14159/180) x=r*cos(v)+(r*cos(b+a*t))*cos(v) y=r*sin(v)+(r*cos(b+a*t))*sin(v) z=r*sin(b+a*t) #MathArt

s=3 (number of lines) r=.5 (torus radius) p=.25 (polygon radius) q=10 (revolutions increment) d=.0075 (degree step) n=0,1,2,3,... a=int(n/s) t= n (mod s) * (2*π/s) u=a*d*(π/180) v=n*d*(π/180) b=n*d*q*(π/180) x=r*cos(v)+(p*cos(b+a*t))*cos(v) y=r*sin(v)+(p*cos(b+a*t))*sin(v) z=p*sin(b+a*t) #MathArt

a=.5*sin(10*u) b=sin(999*u) x=b^2*cos(a+111*u) y=b^2*sin(a+111*u) z=b*cos(999*u) #MathArt

x=b*sin(.5*a)*cos(25*a) y=b*sin(.5*a)*sin(25*a) z=b*cos(.5*a) #MathArt

a=.1*sin(8*u) b=.2*sin(4*u) c=100*sin(u) x=cos(u)+a*cos(u)+b*cos(c*u) y=sin(u)+a*sin(u)+b*sin(c*u) z=a*sin(u)+b*cos(c*u) #MathArt

a=4 b=.2*sin(a*u) c=b*cos(u)+.8*cos(u) d=b*sin(u)+.8*sin(u) e=b*sin(u) n=(360/a/4)*(3.14159/180) f=c*cos(n)+d*sin(n)+.1 g=c*-sin(n)+d*cos(n) h=e s=.001*u t=500*u x=f*cos(s)*cos(t)+g*sin(s)+h*cos(s)*sin(t) y=f*-sin(s)*cos(t)+g*cos(s)+h*-sin(s)*sin(t) z=f*-sin(t)+h*cos(t) #MathArt

x=cos(u) y=sin(u)*cos(v) z=sin(u)*-sin(v) #MathArt

a=.5*sin(1000*u) b=.5*sin(10*u) x=((a+b)*cos(u)+.25)*cos(v) y=(.9*sin(u)+.25) z=((a+b)*cos(u)+.25)*-sin(v) #MathArt

x=(.5*cos(u)+.6)*cos(v) y=.5*sin(u) z=(.5*cos(u)+.6)*-sin(v) #MathArt

One of my ship designs in 3D! Graphed in a chart in Excel. Data was stored and processed in spreadsheets using cell formulas and VBA.

3D surface rendering in Excel. Octahedron with distortion based on multiplying coordinates by a modified distance formula: (x^2+y^2+z^2)^(1/a) where a is between 2 and -2. #MathArt

I figured out how to create 3D surface renderings in Excel! This is a distorted octahedron. #MathArt

USS Enigma. A different class of starship. Comes with a deflector cone additionally designed both as bussard collector and beam weapon. The propulsion branches can create a warp shield. The cone & branches function together with a multiversal quantum drive in addition to warp. #Okudagram #StarTrek

Prompt: Polar coordinates Knot. Made in Excel. #genuary #genuary2026 #genuary10