Carl Sagan once stamped an apple into an ink pad and pressed it onto a sheet of paper. Four small dots appeared. That was his way of showing what a three-dimensional object looks like to a creature who lives on a flat surface, and it set up one of the cleverest explanations of the fourth dimension ever put on television.
It aired in 1980 on Episode 10 of the PBS series “Cosmos.” Sagan opened with a question: “In discussing the large-scale structure of the cosmos, astronomers sometimes say that space is curved. Or that the universe is finite but unbound. Whatever are they talking about?”
The apple in Flatland
He asked viewers to picture a “Flatland,” a two-dimensional plane with no sense of up or down. An apple comes along. The residents of that world would not notice it until it crossed their plane, and even then they would not know what it was. “We see only a fragment as it passes through our plane,” Sagan said. All they have is the evidence of what passed through.
As the apple moves through, its cross-section keeps changing. To someone in Flatland it would look like a set of shapes that shift and rearrange themselves.
The cube that cannot be shown
Then Sagan turned the same logic upward. A tesseract is a cube stretched into a fourth dimension. “I cannot show you a tesseract because I, and you, are trapped in three dimensions,” he said. What he could show was a three-dimensional rendering, the way a cube can be drawn on paper or cast a shadow on it. A tesseract can be represented in 3D space too, though only imperfectly.
Most people have heard that time is the fourth dimension. To pin down an object in the universe, you need its position in space and the moment it exists. Sagan was talking about something harder: a fourth direction that is spatial, like length, width and height.
Where the idea came from
Mathematicians were working on it long before television. Jean le Rond d’Alembert proposed time as a fourth dimension in a 1754 essay. William Rowan Hamilton defined quaternions, an arithmetic of four spatial dimensions, in 1843. Ludwig Schläfli described Euclidean spaces of more than three dimensions in 1852, and his work was not published until 1901.
Charles Howard Hinton helped bring the idea to the public. His 1880 essay “What is the Fourth Dimension?” walked readers step by step from lines to squares to cubes to a “four-dimensional cube.” Hinton also coined the word tesseract, along with “ana” and “kata” for the two extra directions in four-dimensional space.
Hermann Minkowski’s 1908 paper treated time as the fourth dimension of spacetime and gave Einstein’s special and general relativity their geometric basis. Mathematicians were quick to point out that this is a different thing from Schläfli’s four-dimensional Euclidean space. Writing in 1973, H.S.M. Coxeter said little is gained by representing the fourth Euclidean dimension as time, and that the idea, as developed by H.G. Wells in “The Time Machine,” had led some authors into “a serious misconception of the theory of Relativity.”
The spatial version has real uses. Physicists and mathematicians map these spaces to make sense of things they could not otherwise explain. Seeing only three dimensions, Sagan’s lesson goes, does not mean three is all there is.
Sources
- 46 years ago, Carl Sagan beautifully explained the fourth dimension using a sliced apple
- Four-dimensional space – Wikipedia
- Fourth dimension in literature
