Over the last few months, I’ve grown more spiritually, learning and using concepts to better myself, others, and the world we live in.
This next series, we will delve into practices that I’ve been doing for years, have used and tried, and ones that are fairly new for me. I look forward to sharing them with you.
This week, we delve into mathematics and multi dimensions.
Living In Multiple Dimensions
Have you ever wondered what it would be like to live in a world with more than three spatial dimensions? How would you navigate, perceive, and interact with such a reality? We will explore some of the fascinating implications of living in the 11 dimensions, as predicted by some versions of string theory.
String theory is a theoretical framework that attempts to unify all the fundamental forces and particles of nature into a single, consistent picture. One of the key features of string theory is that it requires extra dimensions of space, beyond the familiar three that we experience. Depending on the version of string theory, there could be anywhere from six to 22 extra dimensions, but the most popular one, called M-theory, suggests that there are 11 dimensions in total: three spatial, one temporal, and seven hidden.
The hidden dimensions are not visible to us because they are curled up into tiny shapes called Calabi-Yau manifolds, which are too small to be detected by our current instruments. However, some physicists have speculated that it might be possible to access these dimensions under certain extreme conditions, such as near a black hole or at the beginning of the universe. If that were the case, what would we see and feel in these higher-dimensional realms?
One way to imagine living in the 11 dimensions is to use analogies with lower-dimensional spaces. For example, we can think of a two-dimensional being living on a flat plane, who can only move left-right and up-down, but not forward-backward. To such a being, a three-dimensional object like a sphere would appear as a circle that changes size as it passes through the plane. Similarly, a four-dimensional object like a hypersphere would appear to us as a sphere that changes size as it passes through our three-dimensional space.
However, these analogies have their limitations, because they rely on our ability to visualize higher-dimensional shapes, which is not easy for our brains that are adapted to three dimensions. A more rigorous way to describe living in the 11 dimensions is to use mathematics, specifically geometry and algebra. By using coordinates, vectors, matrices, tensors, and other mathematical tools, we can precisely define the properties and relations of higher-dimensional objects and spaces.
For example, we can define the distance between two points in any dimension by using the Pythagorean theorem. In two dimensions, the distance between (x1,y1) and (x2,y2) is given by:
d = sqrt((x2-x1)^2 + (y2-y1)^2)
In three dimensions, the distance between (x1,y1,z1) and (x2,y2,z2) is given by:
d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
In four dimensions, the distance between (x1,y1,z1,w1) and (x2,y2,z2,w2) is given by:
d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2 + (w2-w1)^2)
And so on for higher dimensions.
By using this formula, we can measure how far apart two points are in any dimension, even if we can’t visualize them.
Another example of using mathematics to describe living in the 11 dimensions is to use matrices to represent rotations. In two dimensions, a rotation around the origin by an angle theta can be represented by a 2×2 matrix:
[cos(theta) -sin(theta)]
[sin(theta) cos(theta)]
This matrix can be multiplied by any 2D vector to obtain its rotated version. In three dimensions, a rotation around an axis by an angle theta can be represented by a 3×3 matrix:
[cos(theta)+u_x^2(1-cos(theta)) u_xu_y(1-cos(theta))-u_zsin(theta) u_xu_z(1-cos(theta))+u_ysin(theta)]
[u_yu_x(1-cos(theta))+u_zsin(theta) cos(theta)+u_y^2(1-cos(theta)) u_yu_z(1-cos(theta))-u_xsin(theta)]
[u_zu_x(1-cos(theta))-u_ysin(theta) u_zu_y(1-cos(theta))+u_xsin(theta) cos(theta)+u_z^2(1-cos(theta))]
where u_x,u_y,u_z are the components of the unit vector along the axis of rotation. This matrix can be multiplied by any 3D vector to obtain its rotated version.
In four dimensions, a rotation around a plane by an angle theta can be represented by a 4×4 matrix:
[cos(theta)+u_0^2(1-cos(theta)) u_0u_1(1-cos(theta))-u_2sin(theta) u_0u_2(1-cos(theta))+u_1sin(theta) u_0u_3(1-cos(theta))]
[u_1u_0(1-cos(theta))+u_2sin(theta) cos(theta)+u_1^2(1-cos(theta)) u_1u_2(1-cos(theta))-u_0sin(theta) u_1u_3(1-cos(theta))]
[u_2u_0(1-cos(theta))-u_1sin(theta) u_2u_1(1-cos(theta))+u_0sin(theta) cos(theta)+u_2^2(1-cos(theta)) u_2u_3(1-cos(theta))]
[u_3u_0(1-cos(theta)) u_3u_1(1-cos(theta)) u_3u_2(1-cos(theta)) cos(theta)+u_3^2(1-cos(theta))]
where u_0,u_1,u_2,u_3 are the components of the unit vector normal to the plane of rotation.
This matrix can be multiplied by any 4D vector to obtain its rotated version. And so on for higher dimensions. By using these matrices, we can perform rotations in any dimension, even if we can’t visualize them.
These are just some examples of how mathematics can help us understand living in the 11 dimensions. Of course, there are many more aspects of higher-dimensional reality that we have not covered, such as curvature, topology, symmetry, and gravity. And, hopefully, this post has given you some insight into the fascinating world of string theory and its implications for our existence.
See you next week!