It’s New Year’s Eve, a day before the new millennium.
A Square is wondering about his smartest grandson’s vivid imagination: he is considering the idea of three dimensions. Square dismisses such silly ideas of course, ignoring his wife’s protests that he is disrespectful.
Suddenly, they feel a presence. Square is confused, the figure that has just appeared is neither a woman, nor a circle; his wife believe it to be a woman.
Square’s wife is asking the stranger to introduce themself and upon realizing the figure is indeed not a woman, rushes to retire to bed.
A Square wants an introduction too - he decides to feel the stranger, even though this is not proper for his status. He is struck by the perfection and utter lack of angles. The stranger isn’t happy about the intrusion and finally explains that he is from Space!
The stranger attempts to explain three dimensions to Square - how he would need an eye on his inside to perceive the third layer. The idea of height is incomprehensible to Square, so the stranger decides to demonstrate:
He is a sphere, he says, and Square sees but a single section of his body - he slower ascends upwards, growing thinner and thinner until he vanishes entirely from Square’s view.
All this demonstration does, is baffle Square. To try again, Sphere is trying to explain three dimensions to Square by using his own shape: a cube. Once he has explained that a cube has 6 sides or insides by Flatland standards, Square is utterly scandalized. In his terror, he attacks!
The attack is easily dodged by Sphere, who simply ascends upwards. He must try again, because we learn that he is only allowed to present his existence to Flatlanders once every 1000 years.
Sphere decides to present his knowledge of Flatland, which clearly he can only have by being from Spaceland: all spaces in Flatland are really planes and to demonstrate that there are no solids, Sphere ascends from above into Square’s cupboard and takes one of his accounting tablets - had the cupboard been a solid structure, he could have only done so via its door, but because it is a plane, he can do so from above or below, leaving the door undisturbed.
Square is shocked but still unconvinced - Sphere has only one weapon left: he touches Square on at and in his stomach.
This backfires horribly - Square is now so terrified that he attacks once more, this time pinning Sphere underneath him.
There is nothing left to do: Sphere is pushing Square upwards, ever higher until they arrive in Spaceland.