Tertium Organum

completely independent
phenomenon or object. The idea of a tree with its branches can never even 
occur to him. 
Altogether, to understand even the most fundamental and simple things of 
our world will be, for the plane being, an infinitely long and difficult process. 
He will have to remodel his ideas of space and time. This must be the first 
step. Nothing can be achieved until this is done. So long as the plane being
visualizes all our universe in time, i.e. refers to time everything that lies on 
both sides of his plane, he will never understand anything. In order to begin 
to understand the 'third dimension', the two-dimensional being living on the 
plane must visualize all his time-concepts 
spatially,
i.e. translate his time into 
space. 
To achieve even an inkling of a right conception of our world, he must 
completely reconstruct all his ideas of the world -
revalue all his values,
re­
examine all his concepts; he must disunite all those concepts which unify and 
bring together those which disconnect and, above all, he must create an 
infinite number of new concepts. 
If we place five fingertips on the plane of the two-dimensional being, this 
will represent for him five separate 
phenomena. 

---

Let us try to imagine the enormous mental evolution the plane being must undergo 
to understand that the five separate phenomena on his plane are the fingertips of the 
hand of a large, active and intelligent being - man. 
It would be extremely interesting to follow, step by step, the road the plane being 
must travel to come to the understanding of our world which, for him, lies in the region 
of the mysterious 
third dimension, 
i.e. partly in the past, partly in the future. In order to 
comprehend the three-dimensional world, the plane being must, first of all, 
cease to be 
two-dimensional, i.e. he must himself become three-dimensional; in other words, he 
must enter into the life interests of a three-dimensional space. If he feels the interests of 
that life he will, by this very fact, draw away from his own plane and will never be able 
to return there. Entering more and more into the orbit of ideas and concepts which 
previously were totally incomprehensible for him, he will no longer be a two­
dimensional being, but will become a three-dimensional one. But for this the plane 
being must really 
be
three-dimensional, i.e. without being aware of it, he must 
possess 
a third dimension. A 
really two-dimensional
being will never become three­
dimensional. In order to 
become
three-dimensional, he must 
be 
three-dimensional. 
Then, in the end, he will be able to get free from the 
illusion 
of the two-dimensionality 
of the world and of himself, and feel the three-dimensional world. 

---

CHAPTER 7 
Impossibility of a mathematical definition of dimensions. Why does mathematics not 
feel dimensions? The entirely conventional character of the designation of dimensions 
by powers. The possibility of representing all the powers on a line. Kant and 
Lobachevsky. The difference between non-Euclidean geometry and metageometry. 
Where should we seek the explanation of the three-dimensionality of the world, if 
Kant's ideas are correct? Are not the three-dimensional conditions of the world to be 
found in our perceiving apparatus, in our mind? 
Now that we have examined the 'relations which our space itself bears within 
it' we must return to the question: