This time we’ll look at an interesting topic: knots in four-dimensional space. Perhaps everyone should already know that due to the extra degree of freedom, all knots in three-dimensional space can be easily untangled in four-dimensional space, so knots are unique products of three-dimensional space. There are no knotting phenomena in four-dimensional space. End of article.

Of course I wouldn’t end this article like that. Although one-dimensional curves cannot form knots, two-dimensional surfaces (two-dimensional ropes?) can form knots! I’ve long heard about four-dimensional knots, but never found specific examples. Claims about Klein bottles are actually incorrect: Klein bottles are non-orientable surfaces. If two-dimensional beings lived on them, they would find themselves “mirrored” after traveling around the bottle once, which could never happen on a sphere. So these two things are fundamentally different (not homeomorphic), like tori and spheres.

Table of Contents:

• Video Recommendation: Knot in 4D

• Episode 1: Three-Dimensional Knots
• Episode 2: Four-Dimensional Knots
• Episode 3: Tube Knots

• Various Types of Holes

• Holes Inside Circles
• Holes Inside Spheres
• Two Linked Spheres?
• Holes Inside (Flat) Tori
• Two Linked Bitori?

• Rotational Knots

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The previous article introduced various types of vehicles in the four-dimensional world, all of which can freely drive and steer on three-dimensional ground surfaces. Since four-dimensional space doesn’t require traffic lights and overpasses, does this mean transportation for four-dimensional beings is extremely convenient? Yes, but four-dimensional traffic still has some minor issues. We know that in three-dimensional space there are two-way roads with median strips or yellow lines separating them, and everyone follows the rule of keeping to the right; in the four-dimensional world’s two-way roads, the median barrier on the three-dimensional road surface must be two-dimensional to separate the roads - drawing a single line no longer serves this purpose. The simplest two-way road would probably look like this:

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Content Overview

In this article, we will focus on curved surface (cell) shapes in four-dimensional space. We’ll first gain deeper understanding of some important solids of revolution from new perspectives (thickening, lofting) that were briefly introduced in the previous article More Geometries (with a list of common solids of revolution!). Then I will introduce some new geometric bodies: double cones, Polytwisters, and four-dimensional spirals.
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This article is a continuation of “Visiting the Four-Dimensional Country”. In that article, I mentioned that 4D urban traffic doesn’t need overpasses because the ground is three-dimensional, and traffic flows in two directions only need to follow skew lines. Now let’s first study what roads in the 4D country actually look like, how cars drive on them, and the analogues of tricycles and bicycles in four-dimensional space. To study general objects in 4D, we can’t use stereographic projection, because stereographic projection essentially studies hyperspheres - we need to project regular polytopes onto hyperspheres. So we’ll start with the cross-section method and projection method. First, let’s look at a straight road with curbs in 3D:
We see that the top view is the clearest, except that you can’t see the height of the curbs. What about 4D roads? Their top view is a three-dimensional cylindrical shape, but should the base be square? Or circular to make the road cylindrical?

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In my previous articles on four-dimensional space, I described a world on a four-dimensional planet. This time, I’ll investigate the day-night and seasonal changes produced by a 4D planet’s rotation and revolution. Let’s assume this planet is approximately a hypersphere. The first thing we need to do is figure out how to describe a point on a hypersphere. This isn’t difficult - similar to spherical coordinates, we can use hyperspherical coordinates to construct a longitude-latitude-like representation. However, since the hypersphere’s surface is three-dimensional, we need three angular measurements.

Before discussing this planet’s rotation, let me briefly introduce rotation of objects in four-dimensional space: The biggest difference from 3D space is the existence of “double rotation.” Rotation is actually an effect unique to two-dimensional space: when rotating the xy plane, the trajectories of all points except the origin are circles in the xy plane. In 3D space, when rotating the xy plane, the projection of spatial points’ trajectories onto the xy plane is also a circle, but note that points on the z-axis don’t move - this is the axis of rotation. In 4D space, when rotating the xy plane, the projection of spatial points’ trajectories onto the xy plane is also a circle, but note that points on the zw coordinate plane don’t move. So we say that 4D space rotation is around a plane. Do you see the issue? The rotating coordinates are only x and y, but we’re concerned with the parts that don’t participate in the rotation, which leads to rotation axes and planes having inconsistent dimensions. Actually, 4D space has “double rotations” where everything except the origin is moving, which forces us to stop focusing on the non-rotating parts: if a rotation combines simultaneous rotations in both the xy and zw planes, this is called double rotation - this is possible because the two planes are absolutely perpendicular, movements in the two directions don’t interfere, and the rotation speeds can differ. It can be proven that all rotations in 4D space are either simple rotations with a stationary rotation plane, or double rotations where all points except the origin are moving, and each double rotation can be decomposed into a composition of two absolutely perpendicular single rotations. This article by Hadroncfy details point movement in double rotation.

Back to planetary rotation: In our universe, due to complex reasons during early formation, planets rotate to varying degrees. For 4D planets, double rotation is the most general state of rotation. We decompose the double rotation into a composition of two absolutely perpendicular single rotations. These two rotations don’t necessarily have equal speeds - we’ll define the faster rotation plane as the xy plane (red circle in the figure below) and the slower plane as the zw plane (red line in the figure below, which is actually also a circle, but appears as a line due to stereographic projection through pole $-w$). This establishes a 4D rectangular coordinate system. Note that the stereographic projection shows the hypersphere’s surface. The marked coordinate points are where the coordinate axes intersect the hypersphere’s surface. The coordinate origin is at the sphere’s center and cannot be represented in stereographic projection.

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(September 2023 update: Added 4DGolf and 4DMiner)

(January-June 2025 update: Added 4D Games, Moena and HoxelDraw)

To further experience the “feeling” of four-dimensional space, creating an interactive program on a computer is ideal. I once dreamed of playing complex 4D modeling software like 3ds Max (unfortunately, it doesn’t exist). Here I’ll organize the four-dimensional programs I’ve played (not necessarily all games) and give some subjective reviews.

The recently popular game in gaming circles, “Miegakure“, is still in development. The game trailer claims to be the first mathematically true 4D game in history, but presents it through the relatively abstract cross-sectional method, with cross-sections chosen perpendicular to the ground. The game renders using tetrahedral cells as basic objects, like triangles in 3D graphics. The trailer shows not only regular hypercuboid room walls, but also 3D ground textures, dodecahedral cell columns and cone combinations, and even four-dimensional trees. Unfortunately, the author has only released a few demos on YouTube, and it’s unclear how long development will take.

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With some basic knowledge of groups, we should now discuss the problem of roots of n-degree equations. Before specifically discussing this problem, we need to introduce some new concepts. Many of these concepts are no longer limited to groups and belong to the scope of abstract algebra. This article is somewhat advanced, but it aims to give readers an intuitive understanding of Galois theory, so general conclusions and proofs will be minimized. Interested readers can refer to Wikipedia or abstract algebra textbooks.

Field Extensions

Let’s first look at another algebraic structure similar to groups—fields.
Simply put, a field is a set with four arithmetic operations defined (addition, subtraction, multiplication, and division), while a group only defines multiplication. We won’t give a strict definition, and this article only deals with number fields, where the four arithmetic operations are the ones we normally use, not other definitions.

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Why do quintic equations have no radical solutions? Galois provided a perfect answer using group theory. For the history and introduction of group theory, I highly recommend this article “Finite Simple Groups: A Century-Long Journey“.
But what specific knowledge do we need to understand Galois theory? Most engineering students haven’t studied group theory or abstract algebra. I plan to write a series of articles, hoping to explain everything starting from zero foundation. Our main thread is understanding Galois theory, though groups have many other applications in mathematics, physics, and chemistry, and I’ll cover some of these other applications.

Table of Contents

• What is a Group?
• Cyclic Groups
• Subgroups
• Cosets and Lagrange’s Theorem
• Symmetric Groups
• Quotient Groups
• Alternating Groups
• Simple Groups

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