in discussing the large scale structure of the cosmos astronomers sometimes say that space is curved or that the universe is finite but unbounded whatever are they talking about let's imagine that we are perfectly flat I mean absolutely flat and that we live appropriately enough in a flat land a land designed and named by Edwin Abbott a Shakespearean scholar who lived in Victoria in England everybody in flatland is of course exceptionally flat we have squares circles triangles and we all Scurry about and we can go into our houses and do our flat business now we have width and length but no height at at all now these little cutouts have some little height but let's ignore that let's imagine that these are absolutely flat that being the case we know us Flatlanders about left right and we know about forward back but we have never heard of up down let us imagine that into flatland hovering above it comes a strange three-dimensional creature which oddly enough looks like an apple and the three-dimensional creature sees an attractive congenial looking Square watches it enter its house and decides in a gesture of interdimensional Amity to say hello hello says the three-dimensional creature how are you I am a visitor from the third dimension well the poor Square looks around his closed house sees no one there and what's more has witnessed a greeting coming from his insides a voice from within he surely is getting a little worried about his sanity the three-dimensional creature is unhappy about being considered a psychological aberration and so he descends to actually enter flatland now a threedimensional creature exists in flatland only partially only a plain a cross-section ction through him can be seen so when the threedimensional creature first reaches flatl it's only the points of contact which can be seen and we'll represent that by stamping the Apple in this ink pad and placing that image in flatland and as the Apple were to descend through sther by flatland we would progressively see higher and higher slices which we can represent by cutting the Apple so the square as time goes on sees a set of objects mysteriously appear from nowhere and inside a closed room and change their shape dramatically his only conclusion could be that he's gone bunkers well the Apple might be a little annoyed at this conclusion and so not such a friendly gesture from Dimension to Dimension makes a contact with the Square from below and sends our flat creature fluttering and spinning above flatland at first the square has no idea what's happening he's terribly confused this is utterly outside his experience but after a while he comes to realize that he is seeing inside closed rooms in flatland he is looking in inside his fellow flat creatures he is seeing flat land from a perspective no one has ever seen it before to his knowledge getting into another dimension provides as an incidental benefit a kind of x-ray vision now our flat creature slowly descends to the surface and his friends Rush up to see him from their point of view he has mysteriously appeared from nowhere he hasn't walked from somewhere else he's come from some other place they say For Heaven's Sake what's happened to you and the poor square has to say well I was in some other Mystic Dimension called up and they will Pat him on his side and comfort him or else they'll ask well show us where is that three di Third Dimension point to it and the poor Square will be unable to comply but maybe more interesting is the other direction in dimensional it what about the fourth dimension now to approach that let's consider a cube we can imagine a cube in the following way you take a line segment and move it at right angles to itself an equal length that makes a square move that square an equal length that right angles to itself and you have a cube now this Cube we understand um casts a shadow [Music] and that shadow we recognize it's you know ordinarily drawn in third grade classrooms as two squares with their vertices connected now if we look at the shadow of a threedimensional object in two Dimensions we see that in this case not all the lines appear equal not all the angles are right angles the three-dimensional object has not been perfectly represented in its projection in two Dimensions but that's part of the cost of losing a dimension in the projection now let's take this threedimensional Cube and project it carry it through a fourth physical Dimension not that way not that way not that way but at right angles to those three directions I can't show you what direction that is but imagine that there is a fourth physical dimension in that case we would generate a four-dimensional hyper Cube which is also called a tesseract I cannot show you a tesseract because I and you are trapped in three dimensions but what I can show you is the shadow in three dimensions of a four-dimensional hyper Cube or TCT this is it and you can see it's two nested cubes all the vertices connected by lines and now the real Tesseract in four dimensions would have all the lines of equal length and all the angles right angles that's not what we see here but that's the penalty of projection so you see while we cannot imagine the world of four dimensions we can certainly think about it perfectly well now imagine a universe just like flatland truly