Dad, Geometry & Space-time

We're at that age aren't we? Our grandparents are gone. My maternal grandmother was the only grandparent I knew, and she died when I first started graduate school.

Now we're losing parents or watching the onset of "mild cognitive decline". My father taught high school math in Vietnam. In the United States, he was just a technician doing quality control on circuit boards, but he managed to teach us geometry, and the high school gave us a diploma to prove it to the other brands like .edu and .com. Now he can't remember what year it is.

I'm up early and thinking about geometry after a night of literature search for a possible job position. My Dad and a possible job position explain all the neurobiology that appears on my stream.

The first thing my Dad taught me was about singularities, about points and dots on a piece of paper.

I woke up thinking about geometry.

I love circles. I wrote a poem about it.

I've drawn circles free-hand.


Did I mention I like to think about the origins of things, like the Universe and "The" Primates?

Let's pretend we are one-dimension (1-D) creatures living in real-time!

We know from our daily living that we can move forward or backwards. But we also know that we move through time. We begin to think of our Universe as a 1-D space-time continuum.

So in the grand scheme of things, we see that the origin was at this dot and the Universe expanded out from the point.

dot singularity

ray singularity It's really difficult to separate time from space. Here we are at the tip of the arrow, looking back in time. It's a straight, linear, 1-D look back. We look at the observable Universe, all of it racing away from us in the past, as a straight line back to the origin, the singularity.

Intuitively, the arrow of space-time looks straight, but what if some mathematician says there is a second dimension?


We are 1-D creatures and it is difficult for us to visualize this. But for mathematician, it's easy, just add another variable


So now we are at the tip of the arrow, but with a new y-axis! The expanding Universe is no longer a straight arrow, (or ray as we say in geometry), but is curved.

seqment singularity

If we look at our Universe, and by definition we are looking back in time, we see that the 2-D space-time arrow is curved.

In this simple model, it actually traces out a half-circle. Ah... my favorite geometric shape!

To us simple 1-D folks, when we look through our telescope and we see the past, we only see a single singularity.

But my weird Geometry says we are actually looking at two symmetrical points of origin.

This symmetry in the singularity is actually us viewing a segment. For a demonstration to kids, we could show them how two singularity points can be connected by a String segment (in baby blue).

string seqment singularity

Adding on dimensions is like an addiction or an addition. I'm really not sure which, but let's add another dimension, the z-direction.

circle area singularity

Now our expanding Universe looks like the curved surface of space-time sphere. (Another favorite geometric shape!)

And our original point singularity has expanded to look like a circle singularity!

We could add another dimension to 4-D, and our circle origin would become the area of the circle. With 5-D, we're looking at the volume of a sphere. How dense would that thing be?!?

Anywho, that's what I was thinking about while half-waking up this morning. I had to share it because sharing is caring.

Or if we don't share, the thought gets lost.

Or am I just instilling in the routine and rituals that will help stay active as mind collapses back to a dark, dense singularity of senescence?

Aging brain show increased reactive astrocytes. Spinal cord injury causes the same. However, this normal response is bad in the CNS. It limits plasticity, which might explain age-related cognitive decline, and very limited to no functional recovery after spinal cord injuries.