Amazing Hinton Cubes that let everyone see the 4D world

Greetings, dear Readers! Today I want to tell you about the remarkable popularizer of mathematics, Charles Howard Hinton, a man who made a huge contribution to the study of objects in the four-dimensional world.
Why, you ask, study them? To begin with, I will limit myself to a quote from his namesake, Howard Lovecraft, whose works I fell in love with not so long ago:

What do we know about this world and the universe? he said. “We have absurdly few sense organs, and our understanding of the surrounding objects is incredibly meager. We see things as we are made to see them, and we are unable to comprehend their absolute essence. With our five miserable senses, we only deceive ourselves, illusoryly imagining that we perceive the entire infinitely complex cosmos – “From the Depths of the Universe (1934)”.

Howard Hinton is a man with a surprisingly interesting biography, but the main merit of his whole life was a series of books devoted to space. Starting with simple examples on a straight line and a plane, Hinton takes the reader to three-dimensional objects, and only then “throws” us into the four-dimensional world with already proven methods.

  Charles Howard Hinton (1853 - 1907).  Born in London, studied mathematics at Oxford and then worked as a teacher.  Main article "What is the fourth dimension" was published in 1883 and brought success and recognition.

Charles Howard Hinton (1853 – 1907). Born in London, studied mathematics at Oxford and then worked as a teacher. The main article “What is the fourth dimension” was published in 1883 and brought success and recognition.

During his life, Hinton developed original methods by which not only professional mathematicians, but also any average person among the growing ranks of his followers, could “see” four-dimensional objects. Finally, Hinton developed special cubes that, with enough effort, helped to visualize hypercubes, or cubes in four dimensions.

They are called “Hinton cubes”. Hinton even coined the official name for the hypercube sweep, the tesseract, which has stuck in the English language. Let’s take a closer look at what the mathematician came up with. One of the ways to visualize objects of higher dimensions is their intersection with objects of lower dimensions. For example, let’s take a 3D cube and pass it through a 2D plane:

It is obvious that we will see one of the faces of the cube, and in three passes we will be able to consider them all, if before that we somehow mark each face. This is exactly the move that Hinton conceives in order to visualize the tesseract. If the hypercube passes through our 3D space, we might perceive its 3D section. The coloring of Hinton’s cubes is based on the matching of four dimensions with four colors: red, yellow, blue and white. When a tesseract point moves across a dimension, the color of that dimension is added to the point’s original color. The final color is the result of mixing. This mixing gives a total of sixteen colors:

Now let’s trace the path of any point of the hypercube in order to understand the principle of forming the colors of its faces. Let’s say we have a “no color” point that moves within the yellow x dimension. The result will be a yellow line segment. Let’s assume that this moving dot starts and ends with the start color and that the inner dots are colored with the measurement color. So we have a line segment that has “color zero” endpoints and yellow interior points:

The line will move within the “red” perpendicular y dimension. The end and start points of the original segment will move in this dimension, leaving a red trace, and the face of the resulting square will become yellow + red = orange!

Now let’s move this square in the third dimension z, which is indicated in blue:

  • Upper face: blue+yellow = green;

  • Edge right: red + blue = purple;

  • Inner faces: red + yellow + blue = brown (according to the dimensions we moved our shapes in!)

Putting together all the possible movements of such points, you can build the following diagram:

Now it remains to implement the methods of intersection with objects of lower dimensions. Passing Hinton cubes through the plane, we get, for example:

But the most interesting will be the visualization of the hypercube itself. We will have four options where the hypercube passes through our space, starting with one cubic face of the Hinton cube and ending with the opposite (similarly colored) face, and all faces will have an additional color depending on the dimension that is not present in the 3D space:

  • If the cube has yellow, white, and red faces, blue will be added.

  • If the cube has yellow, blue, and red faces, white will be added.

  • If it has blue, red and white faces, then yellow will be added.

  • If it has yellow, white and blue faces, then red will be added.

Let’s take the first line as an example: 1. A point without color moves to the right in the white dimension – we get a white segment. 2. The segment moves in the red dimension – we get a square with a pink face (white + red). 3. The square moves in the yellow dimension – we get a cube. 4. Now we direct the hypercube in the fourth dimension (and this is the missing blue color!)

At the exit, the hypercube “turns out” to us with its original side. In fact, you can make Hinton cubes and “play in nature” yourself, spending some effort on this:

Hinton’s cubes were widely advertised in women’s magazines and even used at seances, where they soon acquired mystical significance. Representatives of high society claimed: meditating on Hinton’s cubes, one can catch glimpses of the fourth dimension, and hence the other world of spirits and deceased loved ones. His students spent hours studying these cubes, meditating on them, until they acquired the ability to mentally rearrange and disassemble these cubes through the fourth dimension, receiving a hypercube. It has been argued that one who copes with this mental task is able to achieve the highest state – nirvana.

  Carmelita Chase - Hinton's wife

Carmelita Chase – Hinton’s wife

Hinton himself, after the publication of materials about the fourth dimension, lived a rather measured and successful life until he was arrested (suddenly, Hinton’s father, James Hinton, was a surgeon and supporter of polygamy) for bigamy and was forced to go to Japan, where he worked as a teacher. After moving to North America a few years later, Howard worked at many universities and still enthusiastically told the scientific community and ordinary citizens about the fourth dimension.

I must say, quite successfully. For example, in H. G. Wells’ novel The Time Machine (1895), the Traveler discusses the nature of the fourth dimension in direct text from the story “Incomplete Connection” in Hinton’s second collection (1895). The great Jorge Luis Borges generally included Hinton’s works in his “Babylonian library” and mentioned the Englishman several times in his short but profound stories. Gone in the “fourth dimension” Hinton suddenly – from a brain hemorrhage. After him, a little less than a year later, his wife followed, laying hands on herself. A year after their death in 1908, the German Hermann Minkowski for the first time mathematically rigorously determines the four-dimensional space-time … however, this is a completely different story.

More math in Telegram – “Math is not for everyone”

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