Tertium Organum

The Fourth Dimension,
London, 1912, reprinted Arno Press, New 
York, 1976, pp. 23, 24, 25. 

---

Relation of two- to three-dimensional perception 
Figure 1
 Figure 2 
the movement of the film upwards, we shall have in the film 
a whole world
of moving 
points, whose movements will appear independent to the plane being. 
The plane being will naturally explain these movements as dependent upon one 
another, and the fictitious nature of this movement and its dependence on spirals and 
other lines lying outside his space will never occur to him. 
If we examine the relationship of the plane being to the three-dimensional world we 
shall see that the two-dimensional plane being would find it very difficult to 
understand all the 
complexity of 
the phenomena of our world, as it appears to us. The 
plane being is accustomed to represent to himself too simple a world. 
Taking sections of bodies for bodies, the plane being would compare them only as 
regards their length and their greater or lesser curvature, i.e. 
for him
their greater or 
lesser speed of motion. Such differences as exist 
for us
between the things of our 
world, could not exist for him. The functions of the objects of our world would be 
utterly beyond his understanding; they would be incomprehensible, 'supernatural'. 
Imagine a coin and a candle, both of the same diameter, placed on the plane on 
which the two-dimensional being lives. For the plane being these would be two equal 
circles, i.e. two moving lines, 
absolutely identical;
he would never discover any 
difference between them. The functions which the coin and the candle have 
in our 
world 
would be for him entirely 
terra incognita.
If we try to imagine what a 
tremendous evolution the plane being would have to undergo in order to understand 
the functions of the coin and the candle and the difference between these functions, we 
should understand 
what it is 
that divides the plane world from the three-dimensional 
world. 

---

Before anything else, they are divided because of the utter impossibility - on 
a plane - of even imagining anything like the three-dimensional world with 
all the variety of its functions. 
The properties of the phenomena of the plane world will be extremely 
monotonous; phenomena will be distinguished by the order of their 
appearance, their duration, their periodicity. Bodies and objects of this world 
will be flat and uniform, like 
shadows, 
i.e. like the shadows of completely
different objects, which seem to us alike. Even if the consciousness of a 
plane being could enter into communication with our consciousness, he 
would still be unable ever to understand all the variety and richness of 
phenomena of our world and the variety of functions of our objects. 
Plane beings would be unable to grasp any of our most ordinary concepts. 
It would be very difficult for them to understand that phenomena which are 
the same for them
are actually 
different 
and that, on the other hand, 
phenomena which are quite separate for them are actually parts of one big
phenomenon, or even parts of one object or one being. 
This last would be one of the most difficult things for the plane being to 
understand. If we suppose our two-dimensional being to live on a horizontal 
plane, intersecting the top of a tree, but parallel to the earth, then for him the 
sections of branches will appear each as a