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How to Draw Four Dimensional Figures

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In this article we will see how to draw a two-dimensional representation of a four-dimensional object.

Fig. 1. The number line showing 1D space

Fig. 1. The number line showing 1D space

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. 2. X and Y axes of 2D space

Fig. 2. X and Y axes of 2D space

Fig. 3. One plane of 2D space

Fig. 3. One plane of 2D space

Fig. 3 shows the x,y plane, indicted by a square that contains the x,y axes of 2D space. These axes are 90 degrees 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 2D coordinate system is a single plane.

Fig. 4. The X, Y, and Z axes of 3D space

Fig. 4. The X, Y, and Z axes of 3D space

Fig. 5. The three planes of 3D space.

Fig. 5. The three planes of 3D space.

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

Fig. 6. The four mutually perpendicular axes of 4D space. These can represent three spatial axes and one time axis or four spatial axes x,y,z,w

Fig. 6. The four mutually perpendicular axes of 4D space. These can represent three spatial axes and one time axis or four spatial axes x,y,z,w

Fig. 7. The three 3D planes on their 4D axes.

Fig. 7. The three 3D planes on their 4D axes.

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

Fig 8. The rotation of a 2D figure on a single plane.

Fig 8. The rotation of a 2D figure on a single plane.

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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 3D Figure

By expanding these two equations into six equations and using points indicated by three numbers, we produce a 2D representation of a 3D object. When one plane is rotated the whole figure is rotated by the same amount. By using three different angles of rotation this representation of a three-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)

Fig. 9. A cube is a 3D object.

Fig. 9. A cube is a 3D object.

Using the Equations for the 4D Figure

By expanding these two equations into 12 equations and using points indicated by four numbers, we produce a 2D representation of a 4D object. By rotating any or all of the six planes of the 4D object, the representation of a four-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)

Fig. 10. Tesseract or four dimensional hypercube

Fig. 10. Tesseract or four dimensional hypercube