In this article we will see how to draw a 2- dimensional representation of 4- dimensional objects.

Fig. 1 shows the x-axis or number line. This is a single dimension. Any point on the line is represented by a single number (+x or -x) that indicates its distance from the origin (0).

Fig. 3 shows the x,y plane, indicted by a square that contains the x,y axes of 2-D space. These axes are 90^O to each other.  Any point on the plane is located by two numbers (x,y).  X is the distance from the y-axis to the point.  Y is the distance from the x-axis to the point.  The 2-D coordinate system is a single plane.

Any point in 3-D space is located by 3 numbers (x,y,z).  The 3-D coordinate system consist of 3 planes.  Here these planes are indicated by squares and they are each 90^O to each other.  Because we are viewing the planes at an angle and their image is flatten to the 2-D surface of the page, the squares do not look like squares and the angles do not appear to be 90^O.  However, we are use to seeing squares at an angle and can accept the drawing as representing 3 perpendicular squares.

The 4-D coordinate system consist of 6 planes. This equals all the paired combinations of the axes, xy, xz, xw, yw, zw and yz. That is the number of combinations of n objects taken r at a time = n!/r!(n-r)! = 4!/2!(4-2)! = 24/4 = 6. 

Just as the 3-D planes appear distorted when projected on a 2-D surface, these planes in 4 dimensions are even more distorted when projected on a 2-D surface.   Fig. 7 shows the 2-D projection of the 6 planes that described 4-D space.  Any point in 4-D space is located by 4 numbers (x,y,z,t).  A representation of 4-D space is like a time exposed photo since each 3-D section occurs at a each different instant in time.  This 4-D space is the Minkowski space when the Lorentz transformations are used with this coordinate system.

In analectic geometry there are two combined equations used for rotating all the 2D points in an object by the angle q, on the x,y plane.  These equations are

x’ = x*cos q – y*sin q and

y’ = x*sin q +y*cos q.

Using the equations for the 3-D figure

By expanding these two equations into 6 equations and using points indcated by 3 numbers, produces a 2D representation of a 3D object. When one plane is rotated the whole figure is rotated by the same amount.  By using 3 different angles of rotation this representation of a 3- dimensional object can be viewed from any angle.

Algorithm that produces the 3D effect

XA=X*COS(A1)-Y*SIN(A1):

YA=X*SIN(A1)+Y*COS(A1)

XB=XA*COS(A2)-Z1*SIN(A2)

ZA=XA*SIN(A2)+Z1*COS(A2)

ZB=ZA*COS(A3)-YA*SIN(A3)

YB=ZA*SIN(A3)+YA*COS(A3)

Using the equations for the 4-D figure

By expanding these two equations into 12 equations and using points indcated by 4 numbers, produces a 2D representation of a 4D object. By rotating any or all of the six planes of the 4D object the representation of a 4- dimensional object can be viewed from any angle.

 Algorithm that produces 4D image

 ZA=Z*CQS(A1)-W*SIN(A1)

WA=Z*SIN(A1)+W*COS(A1)

YA=Y*COS(A2)-WA*SIN(A2)

WB=Y*SIN(A2)+WA*COS(A2)

XA=X*COS(A3)-ZA*SIN(A3)

ZB=X*SIN(A3)+ZA*COS(A3)

XB=XA*COS(A4)-WB*SIN(A4)

WC==XA*SIN(A4)+WB*COS(A4)

YB=YA*COS(A5)-ZB*SIN(A5)

ZC=YA*SIN(A5)+ZB*COS(A5) ^:

XC=XB*COS(A6)-YB*SIN(A6)

YC=XB*SIN(A6)+YB*COS(A6)

X2=K*(XC+XC*ZC/800+XC*WC/800)+158:REM adds perspective to x(k=scale)

Y2=0.8*K* (YC+YC*ZC/800+YC*WC/800)+100:REM adds perspective to y(k=scale)

The computer program 4D-CUBE draws the tesseract

This program draws a 2-dimentional representation of a 4-dimentional hypercube.  Each of the 16 points or vertices are indicated by 4 numbers.  One number for the x-axis, the y-axis, the z-axis and the w-axis.  The 32 edges are indicated by drawing a line between two vertices.  Fig. 12 & 13 show the hypercube at different degrees of rotation.  Fig. 14 shows the hypercube with no rotation around any axis.  Because the program has a perspective factor in it fig. 14 appears as 3 connected squares.  Without perspective fig. 14 would appear as a single square just as a 3D cube does.  In GW Basic the height of the pixels is greater than the width.  When figures are viewed on the screen they are taller than they should be. By multiplying the final y by 0.8, these drawings have been adjusted so that the height and the width are proportionally corrected.

4D hypercube and the rotation of each plane

ROTATION OF ZW PLANE 10^O

ROTATION OF YW PLANE 20^O

ROTATION OF XZ PLANE 30^O

ROTATION OF XW PLANE 40^O

ROTATION OF YZ PLANE 50^O

ROTATION OF XY PLANE 60^O

4D hypercube and rotation of planes . . 4D hypercube and rotation of planes

ROTATION OF ZW PLANE 50^{O . . . . . . . .} ROTATION OF ZW PLANE 0^O

ROTATION OF YW PLANE 50^{O . . . . . . .} ROTATION OF YW PLANE 0^O

ROTATION OF XZ PLANE 50^{O . . . . . . . .} ROTATION OF XZ PLANE 0^O

ROTATION OF XW PLANE 0^{O . . . . . . . .} ROTATION OF XW PLANE 0^O

ROTATION OF YZ PLANE 0^{O . . . . . . . .} ROTATION OF YZ PLANE 0^O

ROTATION OF XY PLANE 0^{O . . . . . . . .} ROTATION OF XY PLANE 0^O

The computer program 4D-CUBE in GW Basic

100 CLS:REM 4D-CUBE 110 DIM X(300)

120 DIM Y(300)

130 DIM Z(300)

140 DIM W(300)

150 INPUT "SCALE (suggest 1)";K

160 INPUT "ROTATION OF ZW PLANE IN DEGREES";Al

170 INPUT"ROTATION OF YW PLANE IN DEGREES";A2

180 INPUT "ROTATION OF XZ PLANE INDEGREES";A3

190 INPUT "ROTATION OF XW PLANE IN DEGREES";A4

200 INPUT "ROTATION OF YX PLANE IN DEGREES";A5

210 INPUT "ROTATION OF XY PLANE IN DEGREES";A6

230 A1=A1/57.29577951 converts degrees to radians

240 A2=A2/57.29577951

250 A3=A3/57.29577951

260 A4=A4/ 57,29577951

270 A5=A5/57.29577951

280 A6=A6/57.29577951

290 SCREEN 1,0: CLS:KEY OFF:COLOR 0,1 320 FOR N=I TO 2

330 READ X,Y,Z,W:REM reads data

340 IF X=1.000 THEN 530

350 ZA=Z*CQS(A1)-W*SIN(A1) :REM Algorithm that produces 4D image

360 WA=Z*SIN(A1)+W*COS(A1)

370 YA=Y*COS(A2)-WA*SIN(A2)

380 WB=Y*SIN(A2)+WA*COS(A2)

390 XA=X*COS(A3)-ZA*SIN(A3)

400 ZB=X*SIN(A3)+ZA*COS(A3)

410 XB=XA*COS(A4)-WB*SIN(A4)

420 WC==XA*SIN(A4)+WB*COS(A4)

430 YB=YA*COS(A5)-ZB*SIN(A5)

440 ZC=YA*SIN(A5)+ZB*COS(A5) ^:

450 XC=XB*COS(A6)-YB*SIN(A6)

460 YC=XB*SIN(A6)+YB*COS(A6)

470 if n=1 then 540

480 X2=K*(XC+XC*ZC/800+XC*WC/800)+158:REM adds perspective to x(k=scale)

490 Y2=0.8*K* (YC+YC*ZC/800+YC*WC/800)+100:REM adds perspective to y(k=scale)

500 NEXT N

505 IF W=40 THEN 1000: REM highlights one 3D cube

510 LINE (X1_, Y1)-(X2,Y2),3:REM draws figure

520 GOTO 320

530 END

540 X1=K* (XC+XC*ZC/800+XC*WC/800)+158:REM adds perspective to x(k=scale)

550 Y1=0.8*K*(YC+YC*ZC/800+YC*WC/800)+100:REM adds perspective to y(k=scale)

560 GOTO 500

600 DATA -40,-40,40,-40,40,-40,40,-40

610 DATA -40, -40, 40, 40, 40, -40, 40, 40

620 DATA 40,-40,40,-40,40,40,40,-40

630 DATA 40,-40/40,40,40,40,40,40

640 DATA 40,40,40,-40,-40„40,40,-40

650 DATA 40,40,40,40,-40,40,40,40

660 DATA -40,40,40,-40,-40,-40,40,-40

670 DATA -40 ,40. 40, 40, -40, -40, 40, 40

680 DATA -40,-40,-40,-40,40,-40,-40,-40

690 DATA -40, -40, -40, 40, 40, -40, -40, 40

700 DATA 40,-40,-40,-40,40,40,-40,-40

710 DATA 40,-40,-40, 40, 40, 40,-40, 40

720 DATA 40, 40, -40, -40, -40, 40, -40, -40

730 DATA 40,40,-40,40,-40,40,-40,40

740 DATA -40,40,-40,-40,-40,-40,-40,-40

750 DATA -40,40,-40,40,-40,-40,-40,40

760 DATA -40, -40, 40, -40, -40,-40,-40,-40

770 DATA -40,-40,40,40,-40,-40,-40,40

780 DATA 40, -40, 40, -40, 40, -40, -40, -40

790 DATA 40, -40, 40, 40_, 40, -40, -40, 40

800 DATA 40, 40_,40,-40,40,40,-40,-40

810 DATA 40,40,40,40_,40_,40,-40,40

820 DATA -40,40,40,-40,-40,40,-40,-40

830 DATA -40, 40, 40,40, 40 , 40, -40, 40

840 DATA -40,-40,40,-40,-40,-40,40,40

850 DATA 40,-40,40,-40,40,-40, 40,40

860 DATA 40,40,40,-40,40,40,40,40

870 DATA -40,40,40,-40,-40,40,40,40

880 DATA -40,-40,-40,-40,-40,-40,-40,40

890 DATA 40,-40, -40, -40, 40, -40, -40, 40

900 DATA 40,40,-40,-40,40,40,-40,40

910 DATA -40,40,-40,-40,-40, 40,-40, 40

920 DATA 1000,1000,1000,1000

1000 LINE(X1,Y1)-(X2,Y2),2

1010 GOTO 320

The computer program 4D-Plane

 This program draws a 2-dimentional representation of the six planes of the 4-axes.  Fig. 15 shows the six planes and the coordinates in 4D space.  A layout like this is helpful before drawing any 4-D figure.  By using 6 different angles of rotation this representation of 4- dimensional planes can be viewed from any angle.  When all the angles are at zero we see the x.y plane as a square.  All the other planes are edge on.

Fig. 15 lays out all the points in the 4D coordinate system. These are used in the computer program 4D-PLANE

In fig. 16 to 18 we see 3D planes are drawn with blue lines, while all the planes containing the w-axis are drawn in red.  When we run the computer program the 3D planes are drawn with white lines while all the planes containing the w-axis are drawn in purple.  In GW Basic the height of the pixels is greater than the width.  When figures are viewed on the screen they are taller than they should be. By multiplying the final y by 0.8, these drawings have been adjusted so that the height and the width are proportionally corrected.    

4D planes and the rotation of planes 4D planes and the rotation of planes

ROTATION OF ZW PLANE 0^{O . . . . . . . .} ROTATION OF ZW PLANE 0^O

ROTATION OF YW PLANE 0^{O. . . . . . . .} ROTATION OF YW PLANE 0

ROTATION OF XZ PLANE 0^{O  . . . . . . . .} ROTATION OF XZ PLANE 30^O

ROTATION OF XW PLANE 0^{O  . . . . . . .} ROTATION OF XW PLANE 50^O

ROTATION OF YZ PLANE 0^{O  . . . . . . . .}ROTATION OF YZ PLANE 70^O

ROTATION OF XY PLANE 0^{O . . . . . . . .} ROTATION OF XY PLANE 90^O

4D coordinate planes and the rotation of each plane

ROTATION OF ZW PLANE 10^O

ROTATION OF YW PLANE 20^O

ROTATION OF XZ PLANE 30^O

ROTATION OF XW PLANE 40^O

ROTATION OF YZ PLANE 50^O

ROTATION OF XY PLANE 60^O

^{This figure is complex and hard to visulise. To help in the visualization of this figure in fig. 18a it is separated into 2 parts. First are the 3-d planes with the x,y,z axis. Second are the planes containing the w axis.}

Computer program 4D-PLANE

100 CLS:REM 4D-PLANE

 110 DIM XC300) . 120 DIM Y(300) 130 DIM Z(300) 140 DIM W(300) 150 INPUT "SCALE";K

160 INPUT "ROTATION OF ZW PLANE IN DEGREES";Al

 170 INPUT "ROTATION OF WY PLANE IN DEGREES";A2

 180 INPUT "ROTATION OF XZ PLANE IN DEGREES"; A3

 190 INPUT "ROTATION OF XW PLANE IN DEGREES";A4

 200 INPUT "ROTATION OF YZ PLANE IN DEGREES";A5

210 INPUT "ROTATION OF XY PLANE IN DEGREES";A6

 230 A1=A1/57. 2957795* :REM converts degrees to radians

 240 A2=A2/57.29577951*

 250 A3 =A3/57. 29577951*

 260 A4=A4/57.29577951*

 270 A5=A5/57,29577951*

 280 A6=A6/57.29577951*

 290 SCREEN 1,0:CLS:KEY OFF:COLOR 0,1

 320 FOR.N=1 TO 2

 330 READ X,Y,Z,W

 340 IF X=1000 THEN 530

 350 ZA=Z*COS(A1)-W#SIN(A1):REM algorithms to produce 2D representation of 4D object

 360 WA=Z*SIN(A1)+W*COS(A1)

 370 YA=Y*COS(A2)-WA*SIN(A2)

 380 WB=Y*SIN(A2)+WA*COS(A2)

 390 XA=X*COS(A3)-ZA*SIN(A3)

 400 ZB=X*SIN(A3)+ZA*COS(A3)

410 XB=XA*COS(A4)-WB*SIN(A4)

 420 WC=XA*SIN(A4)+WB*COS(A4)

 430 YB=YA*COS(A5)-ZB*SIN(A5)

 440 ZC=YA*SIN(A5)+ZB*COS<A5)

 450 XC=XB*COS(A6)-YB*SIN(A6)

 460 YC=XB*SIN(A6)+YB*COS(A6)

 470 IF N=l THEN 540

 480 X2=K*(XC+XC*ZC/800+XC*WC/800)+158

 490 Y2=0.8*K*(YC+YC*ZC/800+YC*WC/800)+100

 500 NEXT N

 505 IF W= 40 OR W= -40 THEN 1000

 510 LINE (X1,Y1)-(X2,Y2),3

 520 GOTO 320

 530 END

 540 X1=K*(XC+XC*ZC/800+XC*WC/800)+158

 550 Y1=0.8*K*(YC+YC*ZC/8OO+YC*WC/800)+100

 560 GOTO 500

 600 DATA - 40,- 40„0,0 , 40,- 40,0,0

 610 DATA 40,- 40,0,0,40,40,0,0

 620 DATA 40,40,0,0,-40,40,0,0

 630 DATA - 40,40,0,0,- 40,- 40,0,0

640 DATA - 40 ,0,40, 0, 40, 0., - 40, 0

 650 DATA 40, 0,- 40,0,40,0, 40,0

 660 DATA - 40,0,40,0 _, 40,0,40, 0

 670 DATA - 40, 0, 40, 0, - 40, 0, - 40, 0

 680 DATA 0, - 40, - 40, 0, 0, 40, - 40, 0

 690 DATA 0,40,- 40,0,0,40,40,0

 700 DATA 0,40,40,0,0, - 40, 40, 0

 710 DATA 0,- 40, 40, 0,0,- 40,,- 40,0

 720 DATA - 40,0,0,- 40,40,0,0,-40

 730 DATA 40,0,0,- 40,40,0,0,40

 740 DATA 40,0,0,40,- 40,0,0,40

 750 DATA - 40,0.0,40,- 40,0,0,- 40

 760 DATA 0,- 40,0,- 40,0,40,0,- 40

 770 DATA 0,0, 0, - 40 , 0, 40, 0, 40

 780 DATA 0,40.0, 40,.0,40,0,40

 790 DATA 0, -40, 0, 40, 0, -40,, 6, - 40

 800 DATA 0,0, 40, - 40, 0, 0, - 40, - 40

 810 DATA 0,0,40, 40, 0., 0, 40, - 40

 820 DATA 0,0, 40, 40, 0, 0, - 40, 40

 830 DATA 0,0,- 4,40,0,0,- 40,- 40

 840 DATA 40, 0, 0, 0, - 40, 0, 0.0

850 DATA 0,40,0,0,0,- 40,0,0

 860 DATA 0,0,40,0,0,- 40,0

 870 DATA 0,0,0,40,0,0,0,- 40

 920 DATA 1000,1000,1000,1000

 1000 LINE (X 1, Y1) – (X2, Y2) , 2

 1010 GOTO 320

This program has built in perspective, so that lines more distance from the eye are smaller. To remove this perspective change lines 480, 490, 540, and 550 to;

 480 X2=K*(XC)+158

 490 Y2=K*(YC)+100

 540 X1=K*(XC)+158

 550 Y1=K*(YC)+100

This also true for the previous program the 4D-CUBE. The perspective can be removed from that program by the same lines in it.

An  approximate 4-D drawing can be made by drawing a 3-D drawing of an object twice.  Then connecting the point with lines.  Fig. 20 shows the 4-D coordinate system drawn this way.

Fig. 21 shows the 4-D tetrahedron, cube and octahedron drawn this way.

By understanding these principals you can draw all sorts of 4D figures.  These can be used to study and understand multiple dimensional systems.

Comments

Fourdimensional on October 28, 2017:

Rotation in four-dimensional space.

https://youtu.be/vN9T8CHrGo8

The 5-cell is an analog of the tetrahedron.

https://youtu.be/z_KnvGGwpAo

Tesseract is a four-dimensional hypercube - an analog of a cube.

https://youtu.be/HsecXtfd_xs

The 16-cell is an analog of the octahedron.

https://youtu.be/1-oj34hmO1Q

The 24-cell is one of the regular polytope.

https://youtu.be/w3-TqPXKlVk

The hypersphere is an analog of the sphere.

Mara Alexander from Los Angeles, California on February 27, 2015:

So kewl, this is absolutely awesome. Thank you for sharing

I voted it up

Bennimus - Continuing on October 29, 2014:

I also forgot to mention. The connect-the-dots trick will work on cubes, but you can't use it on tetrahedrons. You will get a "tetrahedronal prism". A proper 5-cell has 5 vertices. Same for octohedrons. It WILL NOT WORK unless you want to end up with a prism.

Bennimus on October 29, 2014:

About the 4D graph. There's a bit of a pattern in 2D representations of higher-dimensional graphs.

Generally, the Z axis goes up. In 2D drawings, Y goes up. So where does Y go? It gets squished down a little bit with the X axis.

The same thing happens when we introduce what's properly known as a W axis. When the W axis enters the scenario, It points up, and it squishes the Z axis alongside the Y and X axes.

All in all it really doesn't matter so long as you have 4 axes, but if you're ever going to do measurements of hypercubes, this is the easiest way to do it.

Rahul on August 01, 2013:

Its really so good, but i know this, can you tell me something about 4-D sphere. My name is Rahul and my age is 14. If you can tell me something about 4-D sphere than please tell me, this is my ID- planetchachi@gmail.com

Thank You Sir

Casper on December 30, 2012:

Excellent!, however, a 4d tetrahedron has only 5 corners!

In your example of a 4d tetrahedron you have made 2 normal tetrahedrons connected to each other bij 90 degree angles into the fourth dimension.

Zack on February 29, 2012:

I could not do any of that! I'm only in 7th grade, and love math, and geometric shapes, but that made no sense. Could you make that clearer, please? my email is lemanski_z@yahoo.com. Thx, Zack.

sve on February 20, 2012:

You made it so simple, it is genius. This is the best explanation of a Fundamental Principle I have seen. Thanks. Keep going.

p johnny joe on November 22, 2011:

sir,it was awesome to see four dimensional figure made by you,when it is going to come in to action,can we make things useful to humanbeings by these figures,100% of the things are made up of 3dimensional,iam a mathematics teacher iam curious to know about it please write in detail to my id johnnyjoe2006@yahoo.com thank you

toxiKrystal on June 06, 2010:

very clear and concise. suprising, considering the amount of usefull informatin here, as well as the complex nature of fourth-dimentional space itself.

i acquired useful knowledge and i am sure there is more to learn here. Bookmarked for sure ^-^

... aren't 4dimentional objects stunningly beautiful ^-^ you definitely aided my task of drawing them by hand.

much thanks

-krys

Mood on May 12, 2010:

Sir you are awesome and passionate, really benefited from this article, thank you and keep up the awesome work coming !