Weekend Open Thread-Introduction to the 4th Dimension

This post will build on something stated on my last WOT post. Take the “East-West” line (x-axis). The “north-south” line (y-axis) cuts it at 90°. The “up-and-down” line (z-axis) cuts both lines at 90° angles. Collectively, these three lines are 90° to each other.

But, what about a line that is simultaneously 90° to these first three lines?

A good starting point, for understanding this, would be through the Flatland story. In 1884, Seeley & Co. published a book written by Edwin A. Abbott: Flatland: A Romance of Many Dimensions.

This is a fictional story involving a square living in a two-dimensional world. This world is nothing but a flat plane. The inhabitants are two-dimensional geometric shapes. The main character is a square named, “A Square.”

Each generation is born with an extra side. So, the children of “A Square” happen to be pentagons. The women are just “lines”. There appears to be a social and economic progression as new generations are born. Circles tend to be in leadership positions.

Workers, soldiers, guards, etc. are “triangles”. The more sides a shape has, the more likely it’ll do “brain” work. Women, having limited rights compared to men, were naturally represented as “lines” in the story.

The author intended to highlight social division, hierarchy, etc., that existed during his time. However, he helped bring the concept of “dimensions” to the public.

A sphere enters Flatland, a square in Flatland sees it as a size-changing circle.

In one scene, “A Square” meets a sphere from the third dimension. This character is from “space land”. When this spherical character enters flatland, “A Square” only sees a cross-section of who he is. He doesn’t see a sphere. Instead, he sees the part of the sphere in the second dimension… A circle.

When the spear first touches Flatland, the square sees a point. That point grows as a circle, stops growing, then shrinks. The circle shrinks to a dot, then disappears. The sphere is completely on the other side of Flatland.

Now, putting us in “A Square’s” place, we could face a similar experience if visited by a fourth-dimensional sphere, a 4-Sphere/hypersphere.

We could project the “sphere through Flatland” scene up one dimension. This time, a 3-dimensional being, us, watch a hypersphere cut through the third dimension. If a hypersphere were to transit from the fourth dimension through our three-dimensional world, we wouldn’t see it in its entirety. We would only see its spherical cross-sections.

As this hypersphere moved into our three-dimensional realm, it would start as a point. That point would expand as a sphere. It would stop expanding, then it would start shrinking. It would shrink down to a point and then it would disappear.

The following videos help capture this concept. The first two videos are short. They provide visualizations of what I talked about above… And more.

Flatland, an introduction:

Beginners Guide to the fourth dimension:

The next two videos go into more detail. Worth the watch. By the time you get done, it would seem like only a few minutes passed by.

Journey into the fourth dimension, part one:

Journey into the fourth dimension, part two:

With the above four, or even the first two, viewed, the following movie would make a little more sense. It’s based on the Flatland story mentioned above. It’s over an hour-long. If you have time, it’s worth the viewing. If you don’t have time, “schedule it” in the “to do” list.

By the time you get done with this movie, you’ll be tempted to click on more videos showing 4-D visualizations:

With that said, enjoy your weekend.

Meanwhile, somewhere in the fourth dimension, a four-dimensional being just read a post and watched videos explaining five-dimensional geometric shapes transiting through the fourth dimension.

Category: Open thread