How to rotate slices of a Rubik’s Cube in python PyOpenGL?

I’m attempting to create a Rubik’s Cube in Python, i have gotten as far as visually representing the cube. Struggling a bit with how to implement rotation.

I guess i’m asking for feedback as to how to go about doing this. I thought at first of, rotating each cubes set of vertices’s, without much luck.

I basically want to select a slice from an array of cube objects (of varying size), perform a rotation and a translation on each object.

import pygame
import random
from pygame.locals import *

from OpenGL.GL import *
from OpenGL.GLU import *

vertices = (
    (1, -1, -1),
    (1, 1, -1),
    (-1, 1, -1),
    (-1, -1, -1),
    (1, -1, 1),
    (1, 1, 1),
    (-1, -1, 1),
    (-1, 1, 1)
    )

edges = (
    (0,1),
    (0,3),
    (0,4),
    (2,1),
    (2,3),
    (2,7),
    (6,3),
    (6,4),
    (6,7),
    (5,1),
    (5,4),
    (5,7)
    )

surfaces = (
    (0,1,2,3),
    (3,2,7,6),
    (6,7,5,4),
    (4,5,1,0),
    (1,5,7,2),
    (4,0,3,6)
    )

colors = (
    (1,0,0), #Red
    (0,1,0), #Green
    (1,0.5,0), #Orange
    (1,1,0), #Yellow
    (1,1,1), #White
    (0,0,1), #Blue
    )

class Cube():
    '''set the vertices edges and surfaces(colored) for a Cube'''
    def __init__(self):
        '''initiate the display to show the cube'''
        pygame.init()
        display = (800,600)
        pygame.display.set_mode(display, DOUBLEBUF|OPENGL)
        glEnable(GL_DEPTH_TEST) 
        gluPerspective(45, (display[0]/display[1]), 0.1, 50.0)

        glTranslatef(1,1, -40)

    def setVertices(self, xmove, ymove, zmove):
        '''set predefined vertices'''
        xValueChange = xmove
        yValueChange = ymove
        zValueChange = zmove

        newVertices = []

        for vert in vertices:
            newVert = []

            newX = vert[0] + xValueChange
            newY = vert[1] + yValueChange
            newZ = vert[2] + zValueChange

            newVert.append(newX)
            newVert.append(newY)
            newVert.append(newZ)

            newVertices.append(newVert)

        return newVertices

    def CreateCube(self, vertices):
        '''create with OpenGL'''
        glBegin(GL_QUADS)
        x = 0
        for surface in surfaces:
            glColor3fv(colors[x])
            x+=1
            for vertex in surface:
                glVertex3fv(vertices[vertex])
        glEnd()

class EntireCube():
    def __init__(self,typeOfCube):
        self.typeOfCube = typeOfCube
        self.NewCube = Cube()

    def createEntireCube(self):
        '''for each dimension x,y,z make a dictionary containing the vertices to be displayed'''
        self.cubeDict = {}
        count = 0
        for x in range(self.typeOfCube):
            for y in range(self.typeOfCube):
                for z in range(self.typeOfCube):
                    self.cubeDict[count] = self.NewCube.setVertices(x*2.1,y*2.1,z*2.1)
                    count += 1

    def mainloop(self):
        '''key events, creates the matrix of cubes'''
        rotateUpKey, rotateDownKey, rotateLeftKey, rotateRightKey = False, False, False, False
        rotationalSensitivity = 2

        while True:
            for event in pygame.event.get():
                if event.type == pygame.QUIT:
                    pygame.quit()
                    quit()
                if event.type == KEYDOWN:
                    if event.key == K_UP:
                        rotateUpKey = True
                    if event.key == K_DOWN:
                        rotateDownKey = True
                    if event.key == K_LEFT:
                        rotateLeftKey = True
                    if event.key == K_RIGHT:
                        rotateRightKey = True

                if event.type == KEYUP:
                    if event.key == K_UP:
                        rotateUpKey = False
                    if event.key == K_DOWN:
                        rotateDownKey = False
                    if event.key == K_LEFT:
                        rotateLeftKey = False
                    if event.key == K_RIGHT:
                        rotateRightKey = False

            if rotateUpKey:
                glRotatef(rotationalSensitivity,-rotationalSensitivity,0,0)
            if rotateDownKey:
                glRotatef(rotationalSensitivity,rotationalSensitivity,0,0)
            if rotateLeftKey:
                glRotatef(rotationalSensitivity,0,-rotationalSensitivity,0)
            if rotateRightKey:
                glRotatef(rotationalSensitivity,0,rotationalSensitivity,0)

            #eventually implement keysbindings to call function to rotate a slice of the matrix created

            # x = glGetDoublev(GL_MODELVIEW_MATRIX)

            glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)

            for eachCube in self.cubeDict:
                self.NewCube.CreateCube(self.cubeDict[eachCube])

            # glPushMatrix()
            # glRotatef(1,3,1,1)
            # glPopMatrix()

            pygame.display.flip()
            pygame.time.wait(10)

def main():
    NewEntireCube = EntireCube(3) #create a 3x3x3 cube
    NewEntireCube.createEntireCube()
    NewEntireCube.mainloop()

if __name__ == '__main__':
    main()
    pygame.quit()
    quit()

I’m hoping someone who knows much more about this can give me some guidance as to how to proceed.

Answer

A rubik’s cube can be organized by an 3 dimensional array of 3x3x3 cubes. It seems to be easy to rotate a slice of the cube, but note if on slice is rotated the positions of the cube change and have to be reorganized. Not only the position changes, also the orientation of the (rotated) single cubes changes.

First of all remove the PyGame and OpenGL initialization form the constructor of the class Cube. That is the wrong place for this. In the following will generate 27 objects of type Cube.

Each cube has to know where it is initially located (self.init_i) and where it is current located after some rotations (self.current_i). This information is encoded in a list with 3 elements, one for each axis. The values are indices of cube in the NxNxN rubik’s cube in range [0, N[.
The orientation of a single cube is encoded in 3 dimensional Rotation matrix (self.rot). The rotation matrix has to be initialized by the identity matrix.

class Cube():
    def __init__(self, id, N, scale):
        self.N = N
        self.scale = scale
        self.init_i = [*id]
        self.current_i = [*id]
        self.rot = [[1 if i==j else 0 for i in range(3)] for j in range(3)]

Create a list of the 27 cubes

cr = range(3)
self.cubes = [Cube((x, y, z), 3, scale) for x in cr for y in cr for z in cr]

If a slice of the rubik’s cube is rotated, then it has to be checked which of the single cubes is affected. This can be done by checking if the slice matches the entry of the rotation axis of the current position.

def isAffected(self, axis, slice, dir):
    return self.current_i[axis] == slice

To rotate a cube, the position and the orientation has to be rotated by 90° around an axis. A 3 dimension rotation matrix consists of 3 direction vectors. A d dimensional vector can be rotated by swapping the coordinates of the vector and inverting the x coordinate of the result for a right rotation and inverting the y coordinate of the result for a left rotation:

rotate right: (x, y) -> (-y, x)
rotate left:  (x, y) -> (y, -x)

Since all the vectors of the rotation matrix are in an axis aligned plane this algorithm can be used to change the orientation and the position of the cube. axis the rotation axis (x=0, y=1, z=2) and dir is the rotation direction (1 is right and -1 left) To rotate the axis vector, 2 components of the vector have to be swapped and one of them inverted.

e.g rotate left around the Y-axis:

(x, y, z) -> (z, y, -x)

When the position is rotated, then the indices have to be swapped. Inverting an index means to map the index i to the index N-1-i:

e.g rotate left around the Y-axis:

(ix, iy, iz) -> (iz, iy, N-1-ix)

Rotation of a single cube:

i, j = (axis+1) % 3, (axis+2) % 3
for k in range(3):
    self.rot[k][i], self.rot[k][j] = -self.rot[k][j]*dir, self.rot[k][i]*dir

self.current_i[i], self.current_i[j] = (
    self.current_i[j] if dir < 0 else self.N - 1 - self.current_i[j],
    self.current_i[i] if dir > 0 else self.N - 1 - self.current_i[i] )

When the cube has to be drawn, then the current position of the cube (self.current_i) and the orientation self.rot can be used to set up 4×4 transformation matrix:

def transformMat(self):
    scaleA = [[s*self.scale for s in a] for a in self.rot]  
    scaleT = [(p-(self.N-1)/2)*2.1*self.scale for p in self.current_i] 
    return [
        *scaleA[0], 0,
        *scaleA[1], 0,
        *scaleA[2], 0,
        *scaleT,    1]

With glPushMatrix respectively glPushMatrix. By glMultMatrix a matrix can be multiplied to the current matrix.
The following function draws a single cube. The parameters angle, axis, slice, dir and it can even apply an animation to the cube, by setting animate=True and setting parameters angle, axis, slice, dir:

def draw(self, col, surf, vert, animate, angle, axis, slice, dir):

    glPushMatrix()
    if animate and self.isAffected(axis, slice, dir):
        glRotatef( angle*dir, *[1 if i==axis else 0 for i in range(3)] )
    glMultMatrixf( self.transformMat() )

    glBegin(GL_QUADS)
    for i in range(len(surf)):
        glColor3fv(colors[i])
        for j in surf[i]:
            glVertex3fv(vertices[j])
    glEnd()

    glPopMatrix()

To draw the cubes, it is sufficient to call the method draw in a loop:

for cube in self.cubes:
    cube.draw(colors, surfaces, vertices, animate, animate_ang, *action)

The implementation of the class Cube works for any NxNxN Rubik’s cube.

See the example program for a 3x3x3 cube. The slices of the cube are rotated to the right by the keys 1 to 9 and to the left by the keys F1 to F9:

Of course the code uses the Legacy OpenGL in regard to your original code. But the method Cube.transformMat sets a general 4×4 model matrix for a single partial cube. Thus it is possible to port this code to modern OpenGL with ease.

import pygame
import random
from pygame.locals import *

from OpenGL.GL import *
from OpenGL.GLU import *

vertices = (
    ( 1, -1, -1), ( 1,  1, -1), (-1,  1, -1), (-1, -1, -1),
    ( 1, -1,  1), ( 1,  1,  1), (-1, -1,  1), (-1,  1,  1)
)
edges = ((0,1),(0,3),(0,4),(2,1),(2,3),(2,7),(6,3),(6,4),(6,7),(5,1),(5,4),(5,7))
surfaces = ((0, 1, 2, 3), (3, 2, 7, 6), (6, 7, 5, 4), (4, 5, 1, 0), (1, 5, 7, 2), (4, 0, 3, 6))
colors = ((1, 0, 0), (0, 1, 0), (1, 0.5, 0), (1, 1, 0), (1, 1, 1), (0, 0, 1))

class Cube():
    def __init__(self, id, N, scale):
        self.N = N
        self.scale = scale
        self.init_i = [*id]
        self.current_i = [*id]
        self.rot = [[1 if i==j else 0 for i in range(3)] for j in range(3)]

    def isAffected(self, axis, slice, dir):
        return self.current_i[axis] == slice

    def update(self, axis, slice, dir):

        if not self.isAffected(axis, slice, dir):
            return

        i, j = (axis+1) % 3, (axis+2) % 3
        for k in range(3):
            self.rot[k][i], self.rot[k][j] = -self.rot[k][j]*dir, self.rot[k][i]*dir

        self.current_i[i], self.current_i[j] = (
            self.current_i[j] if dir < 0 else self.N - 1 - self.current_i[j],
            self.current_i[i] if dir > 0 else self.N - 1 - self.current_i[i] )

    def transformMat(self):
        scaleA = [[s*self.scale for s in a] for a in self.rot]  
        scaleT = [(p-(self.N-1)/2)*2.1*self.scale for p in self.current_i] 
        return [*scaleA[0], 0, *scaleA[1], 0, *scaleA[2], 0, *scaleT, 1]

    def draw(self, col, surf, vert, animate, angle, axis, slice, dir):

        glPushMatrix()
        if animate and self.isAffected(axis, slice, dir):
            glRotatef( angle*dir, *[1 if i==axis else 0 for i in range(3)] )
        glMultMatrixf( self.transformMat() )

        glBegin(GL_QUADS)
        for i in range(len(surf)):
            glColor3fv(colors[i])
            for j in surf[i]:
                glVertex3fv(vertices[j])
        glEnd()

        glPopMatrix()

class EntireCube():
    def __init__(self, N, scale):
        self.N = N
        cr = range(self.N)
        self.cubes = [Cube((x, y, z), self.N, scale) for x in cr for y in cr for z in cr]

    def mainloop(self):

        rot_cube_map  = { K_UP: (-1, 0), K_DOWN: (1, 0), K_LEFT: (0, -1), K_RIGHT: (0, 1)}
        rot_slice_map = {
            K_1: (0, 0, 1), K_2: (0, 1, 1), K_3: (0, 2, 1), K_4: (1, 0, 1), K_5: (1, 1, 1),
            K_6: (1, 2, 1), K_7: (2, 0, 1), K_8: (2, 1, 1), K_9: (2, 2, 1),
            K_F1: (0, 0, -1), K_F2: (0, 1, -1), K_F3: (0, 2, -1), K_F4: (1, 0, -1), K_F5: (1, 1, -1),
            K_F6: (1, 2, -1), K_F7: (2, 0, -1), K_F8: (2, 1, -1), K_F9: (2, 2, -1),
        }  

        ang_x, ang_y, rot_cube = 0, 0, (0, 0)
        animate, animate_ang, animate_speed = False, 0, 5
        action = (0, 0, 0)
        while True:

            for event in pygame.event.get():
                if event.type == pygame.QUIT:
                    pygame.quit()
                    quit()
                if event.type == KEYDOWN:
                    if event.key in rot_cube_map:
                        rot_cube = rot_cube_map[event.key]
                    if not animate and event.key in rot_slice_map:
                        animate, action = True, rot_slice_map[event.key]
                if event.type == KEYUP:
                    if event.key in rot_cube_map:
                        rot_cube = (0, 0)

            ang_x += rot_cube[0]*2
            ang_y += rot_cube[1]*2

            glMatrixMode(GL_MODELVIEW)
            glLoadIdentity()
            glTranslatef(0, 0, -40)
            glRotatef(ang_y, 0, 1, 0)
            glRotatef(ang_x, 1, 0, 0)

            glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)

            if animate:
                if animate_ang >= 90:
                    for cube in self.cubes:
                        cube.update(*action)
                    animate, animate_ang = False, 0

            for cube in self.cubes:
                cube.draw(colors, surfaces, vertices, animate, animate_ang, *action)
            if animate:
                animate_ang += animate_speed

            pygame.display.flip()
            pygame.time.wait(10)

def main():

    pygame.init()
    display = (800,600)
    pygame.display.set_mode(display, DOUBLEBUF|OPENGL)
    glEnable(GL_DEPTH_TEST) 

    glMatrixMode(GL_PROJECTION)
    gluPerspective(45, (display[0]/display[1]), 0.1, 50.0)

    NewEntireCube = EntireCube(3, 1.5) 
    NewEntireCube.mainloop()

if __name__ == '__main__':
    main()
    pygame.quit()
    quit()