Swastika Tiling

An implicit curve: swastika(x,y) = 0

The swastika here is an ancient Buddhist symbol, not the Nazi one.

Composition of swastika(x,y), arctan(x) and tan(x)

(k = 0.25)

Composition of swastika(x,y), arctan(x) and tan(x)

(k = 0.125)

Composition of swastika(x,y), arctan(x) and tan(x)

(k = 0.4)

Composition of swastika(x,y), arcsin(x) and sin(x)

(k = 0.25)

Composition of swastika(x,y), arcsin(x) and sin(x)

(k = 0.4)

Swastika of Swastikas

(k = 1/4, 5/24)

Swastika of Swastikas of Swastikas

(k = 1/4, 1/4, 5/24)

Swastika of Swastikas of Swastikas

(Unicode of 卍: 0x534D)

A Maze of Swastikas

(k = 0.1)

( Mathematical software used: Graph )


Related posts:

• Swastika (卍)

Triple Crescent Moon

Triple Crescent Moon Symbol

(Boundary curves: circles)

Boundary curves: superellipses with exponent 2.5

Boundary curves: superellipses with exponent 4

A monster face!

Hi!

Exponents: 1.5, 2, 2.5, 3, 3.5, 4

( Mathematical software used: gnuplot )

Cross Patty (cross pattée)

max( f(abs(x), abs(y)), f(abs(y), abs(x)) ) > 0

(1) f(x,y) = ax^2 - y + b  (parabola)

a = 0.035, b = 0.85

(2) f(x,y) = x^2 + (y - a)^2 - b^2  (circle)

a = 15.6, b = 14.75

(3) f(x,y) = abs(x)^p + abs(y - a)^p - b^p  (superellipse)

a = 15.6, b = 14.75

p = 2.5, 2.4, 2.3, ...,1.5

p = 4, a = 10, b = 8

(4) f(x,y) = a[exp(-abs(x+b)^p) + exp(-abs(x-b)^p)] - y + c

p = 1.5, a = 1, b = 7, c = 2

p = 3, a = 1, b = 7, c = 2

(5) f(x,y) = a cos(bx)^p - y + c

p = 1, a = -1, b = π/7, c = 2

p = 2, a = 1, b = π/7, c = 2

p = 3, a = -1, b = π/7, c = 2

p = 4, a = 1, b = π/7, c = 2

( Mathematical software used: Graph )

How to Draw the Olympic Rings Mathematically?

Inequalities (Relations) for the Olympic Rings:

1. (Blue Ring)  min(max(R₁(x,y), -y), max(R₁(x,y), y, -R₄(x,y))) < 0
2. (Black Ring)  min(max(R₂(x,y), -y, -R₄(x,y)), max(R₂(x,y), y, -R₅(x,y))) < 0
3. (Red Ring)  min(max(R₃(x,y), -y, -R₅(x,y)), max(R₃(x,y), y)) < 0
4. (Yellow Ring)  min(max(R₄(x,y), -y, -R₁(x,y)), max(R₄(x,y), y, -R₂(x,y))) < 0
5. (Green Ring)  min(max(R₅(x,y), -y, -R₂(x,y)), max(R₅(x,y), y, -R₃(x,y))) < 0

Functions:

• R₁(x,y) = (sqrt((x+2d)² + (y-h)²) - a)² - b²
• R₂(x,y) = (sqrt(x² + (y-h)²) - a)² - b²
• R₃(x,y) = (sqrt((x-2d)² + (y-h)²) - a)² - b²
• R₄(x,y) = (sqrt((x+d)² + (y+h)²) - a)² - b²
• R₅(x,y) = (sqrt((x-d)² + (y+h)²) - a)² - b²

Constants:

• a = 49.25
• b = 4.75
• d = 58.5
• h = 25

Download Examples:

• olympic_rings.zip

( Mathematical softwares used: Graph, gnuplot )

Swastika (卍)

Equation:

(196x² + y² - 1) × (196y² + x² - 1)

× (49x² + 98x + y² + y + 49) × (49x² + 98x + y² - y + 49)

× (49y² - 98y + x² + x + 49) × (49y² - 98y + x² - x + 49)

= 100000000

Note:

The swastika here is an ancient Buddhist symbol, not the Nazi one.
It means good luck, auspiciousness, well-being, and prosperity.

The pronunciation of the character 卍 (wàn) in Chinese is the same
as that of the character 萬 or 万 (means 10,000 or ten thousand).

Rewrite the above equation as

[4(7x)² + y² - 1] × [4(7y)² + x² - 1]

× [(7x)² + 2(7²x) + y² + y + 7²] × [(7x)² + 2(7²x) + y² - y + 7²]

× [(7x)² - 2(7²y) + x² + x + 7²] × [(7x)² - 2(7²y) + x² - x + 7²]

= 10000²,

we see that it is really an equation for good luck.
(It uses the lucky number 7 to produce good luck!)

( Mathematical software used: gnuplot )


Related posts:

• Swastika Tiling