Particularly, the structures of airplane Euclidean geometry may be conducted by way of the rods, e. g. the lattice building, thought of in part 24. unlike ours, the universe of those beings is two-dimensional; yet, like ours, it extends to infinity. of their universe there's room for an unlimited variety of exact squares made of rods, i. e. its quantity (surface) is endless. If those beings say their universe is “plane,” there's experience within the assertion, simply because they suggest that they could practice the structures of airplane Euclidean geometry with their rods. during this connection the person rods constantly signify a similar distance, independently in their place. allow us to reflect on now a moment two-dimensional life, yet this time on a round floor rather than on a airplane. The flat beings with their measuring-rods and different gadgets healthy precisely in this floor and they're not able to depart it. Their entire universe of statement extends completely over the skin of the sector. Are those beings in a position to regard the geometry in their universe as being aircraft geometry and their rods withal because the realisation of “distance”? they can not do that. For in the event that they try and comprehend a directly line, they're going to receive a curve, which we “three-dimensional beings” designate as an excellent circle, i. e. a self-contained line of certain finite size, that are measured up by way of a measuring-rod. equally, this universe has a finite quarter that may be in comparison with the world of a sq. developed with rods. the good attraction due to this attention lies within the reputation of the truth that the universe of those beings is finite and but has no limits. however the spherical-surface beings wouldn't have to move on a world-tour so one can understand that they're now not residing in a Euclidean universe. they could persuade themselves of this on the whole lot in their “world,” supplied they don't use too small a bit of it. ranging from some extent, they draw “straight traces” (arcs of circles as judged in three-d house) of equivalent size in all instructions. they'll name the road becoming a member of the unfastened ends of those traces a “circle. ” For a airplane floor, the ratio of the circumference of a circle to its diameter, either lengths being measured with an identical rod, is, in accordance with Euclidean geometry of the aircraft, equivalent to a continuing price π, that's self sustaining of the diameter of the circle. On their round floor our flat beings may locate for this ratio the worth i. e. a smaller worth than π, the adaptation being the extra substantial, the better is the radius of the circle compared to the radius R of the “world-sphere. ” by way of this relation the round beings can ensure the radius in their universe (“world”), even if just a really small a part of their world-sphere is obtainable for his or her measurements. but when this half is especially small certainly, they'll now not be capable to show that they're on a round “world” and never on a Euclidean aircraft, for a small a part of a round floor differs basically a bit from a bit of a airplane of a similar dimension.