Problem:
Koch Curve Write a program which reads an integer n and draws a Koch curve based on recursive calles of depth n . The Koch curve is well known as a kind of fractals . You can draw a Koch curve in the following algorithm:  Divide a given segment (p1, p2) into three equal segments.  Replace the middle segment by the two sides of an equilateral triangle (s, u, t) of the same length as the segment.  Repeat this procedure recursively for new segments (p1, s), (s, u), (u, t), (t, p2).    You should start (0, 0), (100, 0) as the first segment. Input An integer n is given. Output Print each point (x, y) of the Koch curve. Print a point in a line. You should start the point(0, 0), which is the endpoint of the first segment and end with the point (100, 0), the other endpoint so that you can draw the Koch curve as an unbroken line. Each solution should be given as a decimal with an arbitrary number of fractional digits, and with an absolute error of at most 10-4. Constraints  0 ≤ n ≤ 6  Sample Input 1 1 Sample Output 1 0.00000000 0.00000000
33.33333333 0.00000000
50.00000000 28.86751346
66.66666667 0.00000000
100.00000000 0.00000000 Sample Input 2 2 Sample Output 2 0.00000000 0.00000000
11.11111111 0.00000000
16.66666667 9.62250449
22.22222222 0.00000000
33.33333333 0.00000000
38.88888889 9.62250449
33.33333333 19.24500897
44.44444444 19.24500897
50.00000000 28.86751346
55.55555556 19.24500897
66.66666667 19.24500897
61.11111111 9.62250449
66.66666667 0.00000000
77.77777778 0.00000000
83.33333333 9.62250449
88.88888889 0.00000000
100.00000000 0.00000000 Notes

Solution (Python):


import math
import decimal


def koch(depth, p1, p2):
    if depth == 0:
        return

    s = (decimal.Decimal((2 * p1[0] + 1 * p2[0])) / 3,
         decimal.Decimal((2 * p1[1] + 1 * p2[1])) / 3)
    t = (decimal.Decimal((1 * p1[0] + 2 * p2[0])) / 3,
         decimal.Decimal(((1 * p1[1] + 2 * p2[1]))) / 3)
    u = (
        (t[0] - s[0]) * decimal.Decimal(math.cos(math.radians(60))) -
        (t[1] - s[1]) * decimal.Decimal(math.sin(math.radians(60))) + s[0],
        (t[0] - s[0]) * decimal.Decimal(math.sin(math.radians(60))) +
        (t[1] - s[1]) * decimal.Decimal(math.cos(math.radians(60))) + s[1],
    )

    koch(depth - 1, p1, s)
    print(' '.join(map(lambda x: '{:.8f}'.format(x), s)))
    koch(depth - 1, s, u)
    print(' '.join(map(lambda x: '{:.8f}'.format(x), u)))
    koch(depth - 1, u, t)
    print(' '.join(map(lambda x: '{:.8f}'.format(x), t)))
    koch(depth - 1, t, p2)


if __name__ == "__main__":
    depth = int(input())
    p1 = (decimal.Decimal(0), decimal.Decimal(0))
    p2 = (decimal.Decimal(100), decimal.Decimal(0))
    print(' '.join(map(lambda x: '{:.8f}'.format(x), p1)))
    koch(depth, p1, p2)
    print(' '.join(map(lambda x: '{:.8f}'.format(x), p2)))

Solution (Python):
import math
cos60=math.cos(math.radians(60))
sin60=math.sin(math.radians(60))

def koch(d,p1_x,p2_x,p1_y,p2_y):
    if d==0:
        return 

    s_x=(2*p1_x+1*p2_x)/3
    s_y=(2*p1_y+1*p2_y)/3
    t_x=(1*p1_x+2*p2_x)/3
    t_y=(1*p1_y+2*p2_y)/3
    u_x=(t_x-s_x)*cos60-(t_y-s_y)*sin60+s_x
    u_y=(t_x-s_x)*sin60+(t_y-s_y)*cos60+s_y
    
    koch(d-1,p1_x,s_x,p1_y,s_y)
    print(s_x,s_y)
    koch(d-1,s_x,u_x,s_y,u_y)
    print(u_x,u_y)
    koch(d-1,u_x,t_x,u_y,t_y)
    print(t_x,t_y)
    koch(d-1,t_x,p2_x,t_y,p2_y)
    
d=int(input())
p1_y=0
p1_x=0
p2_x=100
p2_y=0
print(float(p1_x),float(p1_y))
koch(d,p1_x,p2_x,p1_y,p2_y)
print(float(p2_x),float(p2_y))

Solution (Python):
def kochfunc(n1, x1, y1, x2, y2):   
    if n1 > n-1:                      
        return 
    
    root3 = 1.732050807
    sx = (x1*2 + x2)/3 ; sy  = (y1*2 + y2)/3 
    tx = (x1 + x2*2)/3 ; ty  = (y1 + y2*2)/3
    
    vx = tx-sx; vy = ty-sy
    ux = sx + (vx - vy*root3)/2
    uy = sy + (vx*root3 + vy)/2
    
    kochfunc(n1+1, x1, y1, sx, sy)      
    print('{:.8f}'.format(sx), '{:.8f}'.format(sy))
    kochfunc(n1+1, sx, sy, ux, uy)
    print('{:.8f}'.format(ux), '{:.8f}'.format(uy))
    kochfunc(n1+1, ux, uy, tx, ty)
    print('{:.8f}'.format(tx), '{:.8f}'.format(ty))
    kochfunc(n1+1, tx, ty, x2, y2)
    
n = int(input())    
print('{:.8f}'.format(0), '{:.8f}'.format(0))    
kochfunc(0, 0, 0, 100, 0)
print('{:.8f}'.format(100), '{:.8f}'.format(0))

Solution (Python):
def koch(x0,y0, x1,y1, n):
    if n == 0:
        return [(x0,y0)]
    else:
        v = (y0-y1,x1-x0)
        
        
        m = ((x0+x1)/2,(y0+y1)/2)
        c1 = ((2*x0+x1)/3,(2*y0+y1)/3)
        
        c2x = m[0]+v[0]/(2*(3**(1/2)))
        c2y = m[1]+v[1]/(2*(3**(1/2)))
        c2 = (c2x,c2y)
        c3 = ((x0+2*x1)/3,(y0+2*y1)/3)
        return koch(x0,y0, c1[0],c1[1], n-1) \
        + koch(c1[0],c1[1], c2[0],c2[1], n-1) \
        + koch(c2[0],c2[1], c3[0],c3[1], n-1) \
        + koch(c3[0],c3[1], x1,y1, n-1)
def show_vertex(x,y):
    print("%.8f %.8f" % (x,y))

N = int(input())
V = koch(0,0,100,0,N)+[[100,0]]
for x,y in V:
    show_vertex(x,y)

Solution (Python):
import math

def show_vertex(x, y):
  print("%.8f %.8f" % (x, y))

def koch(x0, y0, x1, y1, n):
  if n ==0:
    show_vertex(x0, y0)
  else:
    c1 = [x0 + (x1-x0)/3, y0 + (y1-y0)/3]
    c3 = [x0 + 2*(x1-x0)/3, y0 + 2*(y1-y0)/3]
    c2 = [(x0 + x1)/2 + ((y0-y1)*math.sin(math.pi/3))/3, (y0 + y1)/2 + ((x1-x0)*math.sin(math.pi/3))/3]
    koch(x0,y0, c1[0],c1[1], n-1)
    koch(c1[0],c1[1], c2[0],c2[1], n-1)
    koch(c2[0],c2[1], c3[0],c3[1], n-1)
    koch(c3[0],c3[1], x1,y1, n-1)

N = int(input())
koch(0, 0, 100, 0, N)
show_vertex(100, 0)

Solution (Python):
n= int(input())
def koch(s,t,n):
    if n==0: 
        pass
    else:
        x= (t[0]-s[0])/3  
        y= (t[1]-s[1])/3
        ss= [ s[0]+ x, s[1]+ y ]  
        tt= [ t[0]- x, t[1]- y ]  
        st= [ ss[0]+ x/2- (3**0.5)*y/2 , ss[1]+ (3**0.5)*x/2+ y/2 ] 
        
        koch(s,ss,n-1) 
        print(ss[0],ss[1])  
        koch(ss,st,n-1)
        print(st[0],st[1])
        koch(st,tt,n-1)
        print(tt[0],tt[1])
        koch(tt,t,n-1)
    


print(0,0)
koch([0,0],[100,0],n)
print(100,0)

Solution (Python):
N = int(input())

def solve(x1, y1, x2, y2, n):
    if n == 0:
        return print(x1, y1)
    else:
        solve(x1, y1, (2 * x1 + x2) / 3, (2 * y1 + y2) / 3, n - 1)
        solve((2 * x1 + x2) / 3, (2 * y1 + y2) / 3, (x1 + x2) / 2 + (y1 - y2) * (3 ** (1 / 2)) / 6, (y1 + y2) / 2 + (x2 - x1) * (3 ** (1 / 2)) / 6, n - 1)
        solve((x1 + x2) / 2 + (y1 - y2) * (3 ** (1 / 2)) / 6, (y1 + y2) / 2 + (x2 - x1) * (3 ** (1 / 2)) / 6, (x1 + 2 * x2) / 3, (y1 + 2 * y2) / 3, n - 1)
        solve((x1 + 2 * x2) / 3, (y1 + 2 * y2) / 3, x2, y2, n - 1)

solve(0, 0, 100, 0, N)
print(100, 0)

Solution (Python):
n=int(input())
p=[[0,0]]
r3=1.7320508
def unit_koch(p1,p2,d):
  if d>=n:
    return
  s=[(p1[0]*2+p2[0])/3,(p1[1]*2+p2[1])/3]
  t=[(p1[0]+p2[0]*2)/3,(p1[1]+p2[1]*2)/3]
  v_s_t=[t[0]-s[0],t[1]-s[1]]
  v_s_u=[v_s_t[0]/2-v_s_t[1]*r3/2,v_s_t[1]/2+v_s_t[0]*r3/2]
  u=[s[0]+v_s_u[0],s[1]+v_s_u[1]]
  unit_koch(p1,s,d+1)
  p.append(s)
  unit_koch(s,u,d+1)
  p.append(u)
  unit_koch(u,t,d+1)
  p.append(t)
  unit_koch(t,p2,d+1)
unit_koch([0,0],[100,0],0)
p.append([100,0])
for i in p:
  print(str(i[0])+' '+str(i[1]))

Solution (Python):
import sys
sys.setrecursionlimit(10000000)
MOD = 10 ** 9 + 7
INF = 10 ** 15

N = int(input())
ans = []
const = 2*3**0.5
def korch_curve(Lx,Ly,Rx,Ry,i):
    if i == 0:
        ans.append((Lx,Ly))
        return 
    third_y = (Ly*2 + Ry)/3
    third_x = (Lx*2 + Rx)/3
    korch_curve(Lx,Ly,third_x,third_y,i - 1)
    middle_x = (Lx + Rx)/2 - (Ry - Ly)/const
    middle_y = (Ly + Ry)/2 + (Rx - Lx)/const
    korch_curve(third_x,third_y,middle_x,middle_y,i - 1)
    third_second_x = (Lx + Rx*2)/3
    third_second_y = (Ly + Ry*2)/3
    korch_curve(middle_x,middle_y,third_second_x,third_second_y,i - 1)
    korch_curve(third_second_x,third_second_y,Rx,Ry,i - 1)
korch_curve(0,0,100,0,N)
ans.append((100,0))
for a in ans:
    print(*a)

Solution (Python):
d=[0j,100+0j]
for _ in[0]*int(input()):
 p=[d[0]]
 for i in range(len(d)-1):
  a,b=d[i],d[i+1]
  r=(b-a)/3
  p+=[a+r,a+r+r*(1+3**.5*1j)/2,b-r,b]
 d=p
for e in d:print(e.real,e.imag)

Solution (Python):
def solve(i,x1,x2,y1,y2):
    global n
    if i == n:
        return print(x2,y2)
    X1 = (2*x1+x2)/3
    Y1 = (2*y1+y2)/3
    X3 = (x1+2*x2)/3
    Y3 = (y1+2*y2)/3
    X2 = (X3+X1)/2+3**(0.5)*(Y1-Y3)/2
    Y2 = (3**(0.5)*(X3-X1)+(Y1+Y3))/2
    solve(i+1,x1,X1,y1,Y1)
    solve(i+1,X1,X2,Y1,Y2)
    solve(i+1,X2,X3,Y2,Y3)
    solve(i+1,X3,x2,Y3,y2)

n = int(input())
print(0,0)
solve(0,0,100,0,0)

Solution (Python):
import math
n = int(input())
p1 = (0, 0)
p2 = (100, 0)

def kock(n, p1, p2):
    if n == 0:
        return
    
    sx = 2 / 3 * p1[0] + 1 / 3 * p2[0]
    sy = 2 / 3 * p1[1] + 1 / 3 * p2[1]
    s = (sx, sy)
    
    tx = 1 / 3 * p1[0] + 2 / 3 * p2[0]
    ty = 1 / 3 * p1[1] + 2 / 3 * p2[1]
    t = (tx, ty)
    
    ux = math.cos(math.radians(60)) * (tx-sx) - math.sin(math.radians(60)) * (ty-sy) + sx
    uy = math.sin(math.radians(60)) * (tx-sx) + math.cos(math.radians(60)) * (ty-sy) + sy
    u = (ux, uy)
    
    kock(n-1, p1, s)
    print (*s)
    kock(n-1, s, u)
    print (*u)
    kock(n-1, u, t)
    print (*t)
    kock(n-1, t, p2)
    
print(*p1)
kock(n, p1, p2)
print(*p2)

Solution (Python):
from math import sqrt, sin, cos

def readinput():
    n=int(input())
    return n

def apex3(x1,y1,x2,y2):
    '''正三角形の残りの頂点(x3,y3)を求める
    
    1->2->3の順に反時計回り
    '''
    x3 = 1.0/2.0 *(x2-x1) -sqrt(3.0)/2.0 * (y2-y1) +x1
    y3 = sqrt(3.0)/2.0 * (x2-x1) + 1.0/2.0 * (y2-y1) +y1
    return x3,y3

def koch(x1,y1,x2,y2,l,lmax):
    
    if(l==lmax):
        return

    x3,y3=(x2-x1)/3 +x1, (y2-y1)/3 +y1
    x4,y4=(x2-x1)/3*2 +x1, (y2-y1)/3*2 +y1
    x5,y5 = apex3(x3,y3,x4,y4)

    koch(x1,y1,x3,y3,l+1,lmax)
    print('{:.10f} {:.10f}'.format(x3,y3))
    koch(x3,y3,x5,y5,l+1,lmax)
    print('{:.10f} {:.10f}'.format(x5,y5))
    koch(x5,y5,x4,y4,l+1,lmax)
    print('{:.10f} {:.10f}'.format(x4,y4))
    koch(x4,y4,x2,y2,l+1,lmax)

    return

def main(n):
    x0=0;y0=0
    x1=100;y1=0
    print('{:.10f} {:.10f}'.format(x0,y0))
    koch(x0,y0,x1,y1,0,n)
    print('{:.10f} {:.10f}'.format(x1,y1))


if __name__=='__main__':
    n=readinput()
    main(n)

Solution (Python):
import math

n = int(input())
a = [0, 0]
b = [100, 0]
root3 = math.sqrt(3)


def koch(a, b, n):
    if n == 1:
        s = [(a[0] * 2 + b[0]) / 3, (a[1] * 2 + b[1]) / 3]
        t = [(a[0] + b[0] * 2) / 3, (a[1] + b[1] * 2) / 3]
        u = [(s[0] + t[0] - root3 * t[1] + root3 * s[1]) / 2,
             (s[1] + t[1] + root3 * t[0] - root3 * s[0]) / 2]
        print(*s)
        print(*u)
        print(*t)
        print(*b)

    elif n > 1:
        s = [(a[0] * 2 + b[0]) / 3, (a[1] * 2 + b[1]) / 3]
        t = [(a[0] + b[0] * 2) / 3, (a[1] + b[1] * 2) / 3]
        u = [(s[0] + t[0] - root3 * t[1] + root3 * s[1]) / 2,
             (s[1] + t[1] + root3 * t[0] - root3 * s[0]) / 2]
        koch(a, s, n-1)
        koch(s, u, n-1)
        koch(u, t, n-1)
        koch(t, b, n-1)


print(*a)
koch(a, b, n)
if n == 0:
    print(*b)

Solution (Python):
import math

class KochCurve(object):
	
	x = 0.0
	y = 0.0

	
	

	
	def koch(self, n, a, b):
		if n == 0:
			return 

		s = KochCurve()
		t = KochCurve()
		u = KochCurve()
		th = math.pi * 60.0 / 180.0

		s.x = (2.0 * a.x + 1.0 * b.x) / 3.0
		s.y = (2.0 * a.y + 1.0 * b.y) / 3.0
		t.x = (1.0 * a.x + 2.0 * b.x) / 3.0
		t.y = (1.0 * a.y + 2.0 * b.y) / 3.0
		u.x = (t.x - s.x) * math.cos(th) - (t.y - s.y) * math.sin(th) + s.x
		u.y = (t.x - s.x) * math.sin(th) + (t.y - s.y) * math.cos(th) + s.y
		
		KochCurve().koch(n - 1, a, s)
		msg = "%.8f %.8f" % (s.x, s.y)
		print(msg)
		KochCurve().koch(n - 1, s, u)
		msg = "%.8f %.8f" % (u.x, u.y)
		print(msg)
		KochCurve().koch(n - 1, u, t)
		msg = "%.8f %.8f" % (t.x, t.y)
		print(msg)
		KochCurve().koch(n - 1, t, b)


if __name__ == '__main__':
	a = KochCurve()
	b = KochCurve()

	n = int(input())

	a.x = 0
	a.y = 0
	b.x = 100
	b.y = 0

	msg = "%.8f %.8f" % (a.x, a.y)
	print(msg)
	KochCurve().koch(n, a, b)
	msg = "%.8f %.8f" % (b.x, b.y)
	print(msg)

Solution (Python):
import math


def calc_koch_curve(left, right, i):
    if i == n:
        return
    mid1 = (2*left[0] + right[0]) / 3, (2*left[1] + right[1]) / 3
    mid2 = (left[0] + 2*right[0]) / 3, (left[1] + 2*right[1]) / 3
    mid3_x = (
        (mid2[0] - mid1[0]) * math.cos(math.pi/3)
        - (mid2[1] - mid1[1]) * math.sin(math.pi/3) + mid1[0])
    mid3_y = (
        (mid2[0] - mid1[0]) * math.sin(math.pi/3)
        + (mid2[1] - mid1[1]) * math.cos(math.pi/3) + mid1[1])
    mid3 = (mid3_x, mid3_y)

    calc_koch_curve(left, mid1, i+1)
    print(*mid1)
    calc_koch_curve(mid1, mid3, i+1)
    print(*mid3)
    calc_koch_curve(mid3, mid2, i+1)
    print(*mid2)
    calc_koch_curve(mid2, right, i+1)


def main():
    global C
    global n
    C = [(0, 0)]
    n = int(input())
    print(0., 0.)
    calc_koch_curve((0, 0), (100, 0), 0)
    print(100., 0.)


if __name__ == "__main__":
    main()

Solution (Python):
import math
cos60 = math.cos(60/180*math.pi)
sin60 = math.sin(60/180*math.pi)
class Point():
    def __init__(self, x, y):
        self.x = x
        self.y = y
    def __repr__(self):
        return str(self.x) + " " + str(self.y)
        

def koch(n, p1, p2):
    if(n==0):
        return
    s = Point((2* p1.x + p2.x)/3, (2* p1.y + 1*p2.y)/3)
    
    t = Point((p1.x + 2* p2.x)/3, (p1.y + 2* p2.y)/3)
    
    u = Point((t.x - s.x)* cos60 - (t.y -s.y)* sin60 + s.x,
             (t.x - s.x)* sin60 + (t.y -s.y)* cos60 + s.y)
    
    koch(n-1, p1, s)
    print(s)
    koch(n-1, s, u)
    print(u)
    koch(n-1, u, t)
    print(t)
    koch(n-1, t, p2)

p1 = Point(0,0)
p2 = Point(100,0)
n = int(input())
print(p1)
koch(n, p1, p2)
print(p2)

Solution (Python):
import math
n=int(input())
p1=(0,0)
p2=(100,0)

def koch(n,p1,p2):
    if n==0:
        return

    sx=2/3*p1[0]+1/3*p2[0]
    sy=2/3*p1[1]+1/3*p2[1]
    s=(sx,sy)

    tx=1/3*p1[0]+2/3*p2[0]
    ty=1/3*p1[1]+2/3*p2[1]
    t=(tx,ty)

    ux=math.cos(math.radians(60))*(tx-sx)-math.sin(math.radians(60))*(ty-sy)+sx
    uy=math.sin(math.radians(60))*(tx-sx)+math.cos(math.radians(60))*(ty-sy)+sy
    u=(ux,uy)

    koch(n-1,p1,s)
    print(*s)
    koch(n-1,s,u)
    print(*u)
    koch(n-1,u,t)
    print(*t)
    koch(n-1,t,p2)

print(*p1)
koch(n, p1, p2)
print(*p2)

Solution (Python):
import math
from math import cos, sin, pi

class Point:
    def __init__(self, x=0.0, y=0.0):
        self.x = x
        self.y = y

def koch(n,a,b):
  if n == 0:
    return 0

  th = pi * 60.0 / 180.0

  s, t, u = Point(0.0, 0.0), Point(0.0, 0.0), Point(0.0, 0.0)

  s.x = (2.0 * a.x + 1.0 * b.x) / 3.0
  s.y = (2.0 * a.y + 1.0 * b.y) / 3.0
  t.x = (1.0 * a.x + 2.0 * b.x) / 3.0
  t.y = (1.0 * a.y + 2.0 * b.y) / 3.0
  u.x = (t.x - s.x) * cos(th) - (t.y - s.y) * sin(th) + s.x
  u.y = (t.x - s.x) * sin(th) + (t.y - s.y) * cos(th) + s.y

  koch(n - 1, a, s)
  print(f'{s.x:,.8f} {s.y:,.8f}')

  koch(n - 1, s, u)
  print(f'{u.x:,.8f} {u.y:,.8f}')
  
  koch(n - 1, u, t)
  print(f'{t.x:,.8f} {t.y:,.8f}')

  koch(n - 1, t, b)
    

a = Point(0.0, 0.0)
b = Point(100.0, 0.0)
n = int(input())

print(f'{a.x:,.8f} {a.y:,.8f}')
koch(n,a,b)
print(f'{b.x:,.8f} {b.y:,.8f}')

Solution (Python):
import math

n = int(input())
p0 = (0,0)
p1 = (100,0)

def koch(depth, p0, p1):

	if depth == 0:
		return

	sx = (2*p0[0] + 1*p1[0]) / 3
	sy = (2*p0[1] + 1*p1[1]) / 3
	s = (sx,sy)
	tx = (1*p0[0] + 2*p1[0]) / 3
	ty = (1*p0[1] + 2*p1[1]) / 3
	t = (tx,ty)

	theta = math.radians(60)
	ux = (tx - sx)*math.cos(theta) - (ty - sy)*math.sin(theta) + sx
	uy = math.sin(theta) * (tx - sx) + math.cos(theta) * (ty - sy) + sy
	u = (ux, uy)

	
	koch(depth-1,p0,s)
	print(*s)
	koch(depth-1,s,u)
	print(*u)
	koch(depth-1,u,t)
	print(*t)
	koch(depth-1,t,p1)
	

print(*p0)
koch(n,p0,p1)
print(*p1)

Solution (Python):
import math

x = 0
y = 1
p1 = (0, 0)
p2 = (100, 0)
sin60 = math.sin(math.radians(60))
cos60 = math.cos(math.radians(60))

n = int(input())

def kock(n, p1, p2):
    if n == 0:
        return

    
    s = (0, 0)
    t = (0, 0)
    u = (0, 0)
    s = ((2 * p1[x] + 1 * p2[x]) / 3, (2 * p1[y] + 1 * p2[y]) / 3)
    t = ((1 * p1[x] + 2 * p2[x]) / 3, (1 * p1[y] + 2 * p2[y]) / 3)
    u = ((t[x] - s[x])*cos60 - (t[y] - s[y])*sin60 + s[x], (t[x] - s[x])*sin60 + (t[y] - s[y])*cos60 + s[y])

    kock(n-1, p1, s)
    print ('{:.8f} {:.8f}'.format(s[x], s[y]))
    kock(n-1, s, u)
    print ('{:.8f} {:.8f}'.format(u[x], u[y]))
    kock(n-1, u, t)
    print ('{:.8f} {:.8f}'.format(t[x], t[y]))
    kock(n-1, t, p2)


print ('{:.8f} {:.8f}'.format(p1[x], p1[y]))
kock(n, p1, p2)
print ('{:.8f} {:.8f}'.format(p2[x], p2[y]))

Solution (Python):
import math
n = int(input())

def koch(n, p1, p2):
    if n == 0:
        return

    sx = (2*p1['x'] + 1*p2['x'])/3
    sy = (2*p1['y'] + 1*p2['y'])/3
    tx = (1*p1['x'] + 2*p2['x'])/3
    ty = (1*p1['y'] + 2*p2['y'])/3
    ux = (tx - sx)*math.cos(math.radians(60)) - (ty - sy)*math.sin(math.radians(60)) + sx
    uy = (tx - sx)*math.sin(math.radians(60)) + (ty -sy)*math.cos(math.radians(60)) + sy
    
    s = {'x':sx, 'y':sy}
    t = {'x':tx, 'y':ty}
    u = {'x':ux, 'y':uy}

    koch(n-1,p1,s)
    print('{} {}'.format(s['x'],s['y']))
    koch(n-1, s, u)
    print('{} {}'.format(u['x'],u['y']))
    koch(n-1, u, t)
    print('{} {}'.format(t['x'],t['y']))
    koch(n-1,t,p2)

p1 = {'x':float(0),'y':float(0)}
p2 = {'x':float(100),'y':float(0)}
print('{} {}'.format(p1['x'],p1['y']))
koch(n, p1, p2)
print('{} {}'.format(p2['x'],p2['y']))

Solution (Python):
import sys
from sys import stdin
input = stdin.readline
import math

th = math.pi/3.000000000000000

n = int(input())

p1 = [0.0000000000000000,0.00000000000000000]
p2 = [100.0000000000000,0.00000000000000000]

def koch(n,p1,p2):
    if n == 0:
        return
    s =[0,0]
    t =[0,0]
    u =[0,0]

    s[0] = (2.0 * p1[0] +1.0 * p2[0]) / 3.0
    s[1] = (2.0 * p1[1] +1.0 * p2[1]) / 3.0
    t[0] = (1.0 * p1[0] +2.0 * p2[0]) / 3.0
    t[1] = (1.0 * p1[1] +2.0 * p2[1]) / 3.0
    u[0] = (t[0] - s[0]) * math.cos(th) - (t[1] - s[1]) * math.sin(th) + s[0]
    u[1] = (t[0] - s[0]) * math.sin(th) + (t[1] - s[1]) * math.cos(th) + s[1]
      
    koch(n-1, p1, s)
    print(s[0],s[1])
    koch(n-1, s, u)
    print(u[0],u[1])
    koch(n-1, u, t)
    print(t[0],t[1])
    koch(n-1, t, p2)

print(0,0)
koch(n,p1,p2)
print(100,0)

Solution (Python):
import math


def kock(n, p1, p2):
    if n == 0:
        return

    s = ((p1[0]*2 + p2[0]) / 3, (p1[1]*2 + p2[1]) / 3)
    t = ((p1[0] + p2[0]*2) / 3, (p1[1] + p2[1]*2) / 3)

    rad60 = math.radians(60)
    sin60 = math.sin(rad60)
    cos60 = math.cos(rad60)

    ux = (t[0]-s[0])*cos60 - (t[1]-s[1])*sin60 + s[0]
    uy = (t[0]-s[0])*sin60 + (t[1]-s[1])*cos60 + s[1]
    u = (ux, uy)

    kock(n-1, p1, s)
    print(*s)
    kock(n-1, s, u)
    print(*u)
    kock(n-1, u, t)
    print(*t)
    kock(n-1, t, p2)


n = int(input())
p1 = (0, 0)
p2 = (100, 0)
print(*p1)
kock(n , p1, p2)
print(*p2)

Solution (Python):
import math

def koch(d,p1x,p1y,p2x,p2y):
    if d == 0:
        return 0
    
    sin = math.sin(math.radians(60))
    cos = math.cos(math.radians(60))
    ax = (2 * p1x + 1 * p2x)/3 
    ay = (2 * p1y + 1 * p2y)/3 
    bx = (1 * p1x + 2 * p2x)/3 
    by = (1 * p1y + 2 * p2y)/3
    ux = (bx - ax)* cos - (by - ay)*sin + ax 
    uy = (bx - ax)* sin + (by - ay)*cos + ay

    koch(d-1,p1x,p1y,ax,ay)
    print(ax,ay)
    koch(d-1,ax,ay,ux,uy)
    print(ux,uy)
    koch(d-1,ux,uy,bx,by)
    print(bx,by)
    koch(d-1,bx,by,p2x,p2y)

d = int(input())
print(0,0)
koch(d,0,0,100,0)
print(100,0)

Solution (Python):
import math
def koch(n, p1, p2):
    if n == 0:
        return None
    s = [(p1[0] * 2 + p2[0]) / 3, (p1[1] * 2 + p2[1]) / 3]
    t = [(p1[0] + p2[0] * 2) / 3, (p1[1] + p2[1] * 2) / 3]
    u = [
        (t[0] - s[0]) * math.cos(math.pi / 3) - (t[1] - s[1]) * math.sin(math.pi / 3) + s[0],
        (t[0] - s[0]) * math.sin(math.pi / 3) + (t[1] - s[1]) * math.cos(math.pi / 3) + s[1]
    ]

    koch(n-1, p1, s)
    print(f"{s[0]:.8f} {s[1]:.8f}")
    koch(n-1, s, u)
    print(f"{u[0]:.8f} {u[1]:.8f}")
    koch(n-1, u, t)
    print(f"{t[0]:.8f} {t[1]:.8f}")
    koch(n-1, t, p2)

n = int(input())
print(0, 0)
koch(n, [0.0, 0.0], [100.0, 0.0])
print(100, 0)

Solution (Python):
import math

class Pos:
  def init(x, y):
    self.x = x
    self.y = y

def kock(n, p1, p2):
  if n == 0: return

  s, t = Pos(), Pos()
  s.x = (2 * p1.x + 1 * p2.x) / 3
  s.y = (2 * p1.y + 1 * p2.y) / 3
  t.x = (1 * p1.x + 2 * p2.x) / 3
  t.y = (1 * p1.y + 2 * p2.y) / 3

  cos60 = math.cos(math.radians(60))
  sin60 = math.sin(math.radians(60))

  u = Pos()
  u.x = (t.x - s.x) * cos60 - (t.y - s.y) * sin60 + s.x
  u.y = (t.x - s.x) * sin60 + (t.y - s.y) * cos60 + s.y

  kock(n-1, p1, s)
  print("{:.5f} {:.5f}".format(s.x, s.y))
  kock(n-1, s, u)
  print("{:.5f} {:.5f}".format(u.x, u.y))
  kock(n-1, u, t)
  print("{:.5f} {:.5f}".format(t.x, t.y))
  kock(n-1, t, p2)

n = int(input())

p1, p2 = Pos(), Pos()
p1.x, p1.y = 0.0, 0.0
p2.x, p2.y = 100.0, 0.0

print("{:.5f} {:.5f}".format(p1.x, p1.y))
kock(n, p1, p2)
print("{:.5f} {:.5f}".format(p2.x, p2.y))

Solution (Python):
import math

class point:

    def __init__(self, p):
        self._x = p[0]
        self._y = p[1]

    def disp(self):
        print("{0:f} {1:f}".format(self._x, self._y))

    @property
    def x(self):
        return self._x

    @property
    def y(self):
        return self._y


def get_s(p1, p2):
    p = point((p1.x/3*2 + p2.x/3, p1.y/3*2 + p2.y/3))
    return p


def get_t(p1, p2):
    p = point((p1.x/3 + p2.x/3*2, p1.y/3 + p2.y/3*2))
    return p


def get_u(s, t):
    x = (t.x - s.x)*math.cos(math.radians(60)) - (t.y - s.y)*math.sin(math.radians(60)) + s.x
    y = (t.x - s.x)*math.sin(math.radians(60)) + (t.y - s.y)*math.cos(math.radians(60)) + s.y
    return point((x, y))


def kock(n, p1, p2):
    if n == 0:
        return
    else:
        s = get_s(p1, p2)
        t = get_t(p1, p2)
        u = get_u(s, t)

        kock(n-1, p1, s)
        s.disp()
        kock(n-1, s, u)
        u.disp()
        kock(n-1, u, t)
        t.disp()
        kock(n-1, t, p2)

        return


if __name__ == '__main__':

    n = int(input())

    p1 = point((0, 0))
    p2 = point((100, 0))

    p1.disp()
    kock(n, p1, p2)
    p2.disp()

Solution (Python):
n = int(input())
p1, p2 = (0, 0), (100, 0)
ans_ls = [p1, p2]

def koch(d, p1, p2):
    '''
    d...現在の深さ 0スタートにする
    p1,p2...端の座標。ここから途中の座標を計算しlistにappendする
    '''
    
    if d == n:
        return
    
    s = (2/3 * p1[0] + 1/3 * p2[0], 2/3 * p1[1] + 1/3 * p2[1])
    t = (1/3 * p1[0] + 2/3 * p2[0], 1/3 * p1[1] + 2/3 * p2[1])

    
    
    R = [[1/2, -0.86602540378], [0.86602540378, 1/2]]
    u = (R[0][0] * (t[0] - s[0]) + R[0][1] * (t[1] - s[1]) + s[0],
         R[1][0] * (t[0] - s[0]) + R[1][1] * (t[1] - s[1]) + s[1])
    ans_ls.extend([s, u, t])  

    
    koch(d+1, p1, s)
    print(*s)
    koch(d+1, s, u)
    print(*u)
    koch(d+1, u, t)
    print(*t)
    koch(d+1, t, p2)

print(*p1)
koch(0, p1, p2)
print(*p2)

Solution (Python):
import math

def Koch(Px, Py, Qx, Qy, n) :
    if n == 0 :
        return
    else :
        Ax = (2 * Px + Qx) / 3
        Ay = (2 * Py + Qy) / 3
        Bx = (Px + 2 * Qx) / 3
        By = (Py + 2 * Qy) / 3
        Cx = Ax + (Bx - Ax) * math.cos(math.pi/3) - (By - Ay) * math.sin(math.pi/3) 
        Cy = Ay + (Bx - Ax) * math.sin(math.pi/3) + (By - Ay) * math.cos(math.pi/3)
        
        Koch(Px, Py, Ax, Ay, n-1)
        print(Ax, Ay)
        Koch(Ax, Ay, Cx, Cy, n-1)
        print(Cx, Cy)
        Koch(Cx, Cy, Bx, By, n-1)
        print(Bx, By)
        Koch(Bx, By, Qx, Qy, n-1)

n = int(input())
print(0, 0.00000000)
Koch(0.00000000, 0.00000000, 100.00000000, 0.00000000, n)
print(100.00000000, 0.00000000)

Solution (Python):
from math import sqrt
q3 = sqrt(3)
fmt = "%.8f %.8f"
def koch(x0, y0, x1, y1, c):
    if c == n:
        return (fmt % (x0, y0),)
    xp = (x0*2 + x1) / 3
    yp = (y0*2 + y1) / 3

    xq = (x0 + x1*2) / 3
    yq = (y0 + y1*2) / 3

    dx = (x1 - x0) / 6
    dy = (y1 - y0) / 6

    xr = xp + (dx - dy * q3)
    yr = yp + (dx * q3 + dy)

    return koch(x0, y0, xp, yp, c+1) + koch(xp, yp, xr, yr, c+1) + koch(xr, yr, xq, yq, c+1) + koch(xq, yq, x1, y1, c+1)
n = int(input())
print(*koch(0, 0, 100, 0, 0)+(fmt % (100, 0),), sep='\n')

Solution (Python):
import math
def Koch(tops):
    n = len(tops)
    new_tops = [tops[0]]
    for i in range(n-1):
        b = [0,0]
        a = [2*tops[i][0]/3 + tops[i+1][0]/3, 2*tops[i][1]/3 + tops[i+1][1]/3]
        c = [tops[i][0]/3 + 2*tops[i+1][0]/3, tops[i][1]/3 + 2*tops[i+1][1]/3]
        b[0] = (c[0]-a[0])*math.cos(math.radians(60)) - (c[1]-a[1])*math.sin(math.radians(60)) + a[0]
        b[1] = (c[0]-a[0])*math.sin(math.radians(60)) + (c[1]-a[1])*math.cos(math.radians(60)) + a[1]
        new_tops.append(a)
        new_tops.append(b)
        new_tops.append(c)
        new_tops.append(tops[i+1])
    return new_tops

n = int(input())
tops = [[0.,0.],[100.,0.]]
for i in range(n):
    tops = Koch(tops)

for j in range(len(tops)):
    print(*tops[j], sep=' ')

Solution (Python):
import math

n = int(input())

point = [(0, 0), (100, 0)]

for i in range(n):
    len_point = len(point)
    tmp = []
    for j in range(len_point - 1):
        x1, y1 = point[j]
        x2, y2 = point[j+1]
        sx, sy = (2 * x1 + x2) / 3, (2 * y1 + y2) / 3
        tx, ty = (x1 + 2 * x2) / 3, (y1 + 2 * y2) / 3
        ux = (tx - sx) * math.cos(math.pi / 3) - (ty - sy) * math.sin(math.pi / 3) + sx
        uy = (tx - sx) * math.sin(math.pi / 3) + (ty - sy) * math.cos(math.pi / 3) + sy
        tmp.append((x1, y1))
        tmp.append((sx, sy))
        tmp.append((ux, uy))
        tmp.append((tx, ty)) 
    tmp.append((100, 0))
    point = tmp

for x, y in point:
    print(f"{x:.8f} {y:.8f}")

Solution (Python):
def next_math(n):
    turn60 = 0.5 + 3 ** 0.5 * 0.5j
    n_next = [n[0]]
    for i in range(1, len(n)):
        s = (n[i - 1] * 2 + n[i]) / 3
        t = (n[i - 1] + n[i] * 2) / 3
        u = (t - s) * turn60 + s
        n_next +=[s, u, t, n[i]]
    return n_next

p1 = 0 + 0j
p2 = 100 + 0j
math = [p1, p2]

n = int(input())
for i in range(n):
    math = next_math(math)

for z in math:
    print("{:.8f} {:.8f}".format(z.real, z.imag))

Solution (Python):
n=int(input())
p1=[float(0),float(0)]
p2=[float(100),float(0)]
cos=0.5
sin=0.866025404
def koh(d,p1,p2):
    if d==0:return True
    sx = (2 * p1[0] + p2[0]) / 3
    sy = (2 * p1[1] + p2[1]) / 3
    tx = (p1[0] + 2 * p2[0]) / 3
    ty = (p1[1] + 2 * p2[1]) / 3
    ux=(tx-sx)*cos-(ty-sy)*sin+sx
    uy=(tx-sx)*sin+(ty-sy)*cos+sy

    koh(d-1,p1,[sx,sy])
    print(sx, sy)
    koh(d-1,[sx,sy],[ux,uy])
    print(ux, uy)
    koh(d-1,[ux,uy],[tx,ty])
    print(tx,ty)
    koh(d-1,[tx,ty],p2)
print(p1[0],p1[1])
koh(n,p1,p2)
print(p2[0],p2[1])

Solution (Python):
import math

n = int(input())

p1 = [0.0, 0.0]
p2 = [100.0, 0.0]

def rad(r) :
    return math.radians(r)

def sin(s) :
    return math.sin(s)
    
def cos(s) :
    return math.cos(s)

def Koch(n, p1, p2) :
    if n == 0 :
        return 0
    else :
        s = [0, 0]
        t = [0, 0]
        u = [0, 0]
        
        s[0] = (p2[0] - p1[0]) / 3 + p1[0]
        s[1] = (p2[1] - p1[1]) / 3 + p1[1]
        t[0] = (p2[0] - p1[0]) / 3 * 2 + p1[0]
        t[1] = (p2[1] - p1[1]) / 3 * 2 + p1[1]
        
        u[0] = (t[0] - s[0]) * cos(rad(60)) - (t[1] - s[1]) * sin(rad(60)) + s[0]
        u[1] = (t[0] - s[0]) * sin(rad(60)) + (t[1] - s[1]) * cos(rad(60)) + s[1]
        
        Koch(n - 1, p1, s)
        pri(s[0], s[1])
        
        Koch(n - 1, s, u)
        pri(u[0], u[1])
        
        Koch(n - 1, u, t)
        pri(t[0], t[1])
        
        Koch(n - 1, t, p2)
        
def pri(x, y) :
    print(f"{x:.8f} {y:.8f}")
    
print(p1[0], p1[1])
Koch(n, p1, p2)
print(p2[0], p2[1])

Solution (Python):
import math
n=int(input())

def add_v(l1,l2):
    n=len(l1)
    l=[0]*n
    for i in range(n):
        l[i]=l1[i]+l2[i]
    return l

def scar_multiple_v(scalar,l):
    n=len(l)
    ans=[0]*n
    for i in range(n):
        ans[i]=l[i]*scalar
    return ans

def mat_multi_v(mat,l):
    n=len(l)
    ans=[0]*n
    for i in range(n):
        for j in range(n):
            ans[i]+=mat[i][j]*l[j]
    return ans

def make_points(tail,head):
    ans=[[0]*2 for _ in range(5)]
    ans[0]=tuple(tail)
    ans[1]=tuple(add_v(tail,scar_multiple_v(1/3,add_v(head,scar_multiple_v(-1,tail)))))

    v=scar_multiple_v(1/3,add_v(head,scar_multiple_v(-1,tail)))
    mat=[[1+math.cos(math.radians(60)),-math.sin(math.radians(60))],\
         [math.sin(math.radians(60))  ,1+math.cos(math.radians(60))]]
    ans[2]=tuple(add_v(tail,mat_multi_v(mat,v)))
    ans[3]=tuple(add_v(tail,scar_multiple_v(2/3,add_v(head,scar_multiple_v(-1,tail)))))
    ans[4]=tuple(head)
    return tuple(ans)

def make_1_coch_curve(iter,tail,head):
    points=make_points(tail,head)
    if iter == n-1:
        for i in range(len(points)-1):
            print("{} {}".format(points[i][0],points[i][1]))
            
    else:
        for i in range(4):
            make_1_coch_curve(iter+1,points[i],points[i+1])
            
if n == 0:
    print("{} {}\n{} {}".format(0,0,100,0))
else:
    make_1_coch_curve(0,[0,0],[100,0])
    print("{} {}".format(100,0))

Solution (Python):
from math import cos, sin, pi

def show(p):
    print(f'{p.real:.08} {p.imag:.08}')

def rotate(p, theta):
    return p*complex(cos(theta), sin(theta))

def go(n, p1, p2):
    if n == 0:
        return
    s = (p2 - p1)/3 + p1
    t = 2*s - p1
    u = rotate(t - s, pi/3) + s
    go(n - 1, p1, s)
    show(s)
    go(n - 1, s, u)
    show(u)
    go(n - 1, u, t)
    show(t)
    go(n - 1, t, p2)


p1 = complex(0, 0)
p2 = complex(100, 0)
show(p1)
go(int(input()), p1, p2)
show(p2)

Solution (Python):
import math
def koch(d, p1, p2):
    if d == 0:
        return

    sx = (2*p1[0]+1*p2[0])/3
    sy = (2*p1[1]+1*p2[1])/3
    s = (sx, sy)
    tx = (1*p1[0]+2*p2[0])/3
    ty = (1*p1[1]+2*p2[1])/3
    t = (tx, ty)
    rad = math.radians(60)
    ux = (t[0]-s[0])*math.cos(rad)-(t[1]-s[1])*math.sin(rad)+s[0]
    uy = (t[0]-s[0])*math.sin(rad)+(t[1]-s[1])*math.cos(rad)+s[1]
    u = (ux, uy)

    koch(d-1, p1, s)
    print(s[0], s[1])
    koch(d-1, s, u)
    print(u[0], u[1])
    koch(d-1, u, t)
    print(t[0], t[1])
    koch(d-1, t, p2)

d = int(input())
p1 = (0, 0)
p2 = (100, 0)
print(p1[0], p1[1])
koch(d, p1, p2)
print(p2[0], p2[1])

Solution (Python):
from math import sin, cos, radians

N = int(input())

p1 = (0, 0)
p2 = (100, 0)
th = radians(60)

def koch(n, a, b):
    global th

    if n == 0:
        return
    
    s = ((2 * a[0] + 1 * b[0]) / 3, (2 * a[1] + 1 * b[1]) / 3)
    t = ((1 * a[0] + 2 * b[0]) / 3, (1 * a[1] + 2 * b[1]) / 3)
    u = ((t[0] - s[0]) * cos(th) - (t[1] - s[1]) * sin(th) + s[0],
         (t[0] - s[0]) * sin(th) + (t[1] - s[1]) * cos(th) + s[1])

    koch(n - 1, a, s)
    print(*s)
    koch(n - 1, s, u)
    print(*u)
    koch(n - 1, u, t)
    print(*t)
    koch(n - 1, t, b)

print(*p1)
koch(N, p1, p2)
print(*p2)