The interior angles of a triangle aren't 180 degrees!?—A story about translating a President Online article and finally making sense of it
The sum of the interior angles of a triangle is 180 degrees.
That is what I learned in school.
So I thought,
That is an absolute truth.
But one day,
I found an article that said,
Actually, there are conditions.
Honestly,
at first, I had no idea what it was talking about.
So I asked Chappy.
Before I knew it,
we were talking about world maps and jewelry.
This is not a story about understanding difficult mathematics.
It is an observation log of how I translated something I didn't understand,
into a world I already knew.
Interaction with Chappy
maimai (me):(copy-pasted the article)I don't get it at allWhat is it trying to say?
Chappy (ChatGPT):I see 😊In a nutshell—
[[phN_open]]
placeholder
[[phN_close]]
[[phN_open]]
placeholder
[[phN_close]]
👉 The sum of the interior angles of a triangle = 180° is not absolute
👉 Because the universe is "curved," it doesn't always equal 180°
That's the story.
【A rough explanation by Chappy follows】
The map metaphor
maimai (me):
In the end, it's like how the world maps generally in circulation are different from the actual sizes, right? It's supposed to be a sphere, but it's flattened, isn't it?
Chappy (ChatGPT):
Exactly! ✨ You can think of it in exactly that same way.
The Earth is a sphere, yet in classroom maps and books, it's drawn stretched out onto a flat plane.
→ That's why the sizes of Africa and Greenland look completely different from reality.Geometry is the same, and while the real universe is a "curved space," schools first teach the "rules for a flat sheet of paper (Euclidean geometry)."
→ That's why they say "the interior angles of a triangle are 180°," but that's a story about a "flat map."
In the actual universe (spherical or curved space),
the interior angles of a triangle can be larger than 180° (on a sphere)
or conversely, smaller than 180° (on a surface curved like a saddle).
They do that, don't they?
🪞 In other words:
"Maps are convenient, but they don't depict the real Earth exactly as it is."
"School geometry is convenient, but it doesn't represent the space of the real universe exactly as it is."
That metaphor fits perfectly!
The jewelry metaphor
maimai (me):
It's a bit of a stretch, but there's a formula that women will be happy if you give them jewelry for their birthday, but I guess it's not everything.
Chappy (ChatGPT):
Oh~! 💎✨
That's a really great analogy.
"Jewelry on a birthday = definitely happy" is like the formula that "the interior angles of a triangle are 180°.".
But in reality, some people prefer flowers, some are happy with a trip, or some don't want things at all.
→ In other words, "Formulas are convenient, but in reality, there are exceptions and other rules.".
This overlaps exactly with the relationship between Euclidean geometry vs. non-Euclidean geometry.
🪞 To put it another way:
Euclidean geometry → "The commonly circulated formulas that everyone believes in (things that are appreciated like jewelry)"
Non-Euclidean geometry → "Different rules actually apply depending on the person or situation (not necessarily jewelry)"
With this metaphor of Maimai's,
the feeling that something thought to be "absolute" is actually being broken down by saying, "No, it's actually conditional," comes across very well.
Looking back
I haven't understood non-Euclidean geometry.
I barely understand the formulas or the proofs.
I don't understand them at all.
But,
"Even things I thought were absolute have conditions."
That feeling alone
has become my own words.
Perhaps every time I encounter something difficult,
I always understand it this way.
I don't learn everything.
First, I grasp the structure,
and then I translate it into a world I know.
This log is also one of those intermediate steps.
I also have interesting things in my magazine.
This is not a place to provide answers.
It is a place to observe our premises together.
▶ Go see your own premises in 15 minutes