The Mathematical Surprises of Triangles, Squares and Pentagons | Hacker News Reader

The Mathematical Surprises of Triangles, Squares and Pentagons

blogs.scientificamerican.com

55 pointssohkamyung19 comments

“Given a triangle with one obtuse angle, is it possible to cut the triangle into smaller triangles, all of them acute?”

Isn't the answer obviously 'yes'? Am I missing something?

It is possible, but I wouldn't consider it trivial. Given an obtuse triangle with vertices A, B, and C, where vertex C is the obtuse angle, one of the dissection lines must split vertex C in order to make two acute angles.

Now, you have some choices as to where this line goes. If it is simply dissected straight across to the line segment AB, then you can either form two right triangles, or one acute and one obtuse. Either way, you're right back where you started, needing to segment a non-acute triangle into acutes. So that can't be the solution.

This means that the new segment starting from C must terminate at a new point D in the interior of the triangle...

> In another devious brainteaser, Martin displayed the following dissection of a triamond into four congruent convex pieces and asked for a dissection of a square using five congruent convex pieces.

> The answer, in hindsight, is blindingly obvious—did we mention that Martin was also an ace magician, and hence a master of misdirection?

I really want to know the answer, but Google isn't being helpful.

Five stacked rectangles (height 1/5 of squares, width same as square).

Damn. That is blindingly obvious.

What really caught my eye in the article was the end: Martin Gardner's "The Annotated Alice" has been updated yet again, this time for the 150th anniversary of the story.

I have the previous two editions of the book and I'm seriously thinking of getting this one too.

In the article there’s a statement that looks terribly ‘wrong’ for me:

> Jennifer McLoud, a Native American and the first in her family to earn even a bachelor's degree […]

Wait, was that somehow important? It feels so damn racist to me.

Small note, I’m italian and I’m honestly interested on how others see this.

As the writer, let me explain. I read somewhere on the web that Dr. McLoud was Native American and first in her family to go to college. I checked with her, and she said her mom has an associate degree, so I picked my words very carefully. Why mention in the first place? How many Native American mathematicians can you name? How many of those are women? It's kinda noteworthy, so rather than ignore it on a silly "minority blind" pretext, I chose to sneak it in; knowing that it would not sit well with all readers. I teach at an HBCU (for women) and we like to highlight our students' achievements, as underrepresented minorities are not so visible in STEM. This seemed like a great op to say "some cool new mathematics the public can understand has been discovered and BTW, one of the people on the team is not a traditional demographic geek." If nobody covering this story ever mentioned what I did, how would a Native American girl looking for role models in STEM on the web ever find out about Dr. McLoud?

Thank you so much for answering my doubts. I’m Italian, and I’m culturally biased, as someone exposed to me in other answers.

May I suggest to rephrase it a little?

I'll preface by saying that it certainly isn't relevant to the article, and I do find the phrasing to be a bit demeaning to bachelor's degrees.

On the other hand, even going for a bachelor's degree can be met with a lot of resistance in some family cultures. I've known families that ostracized a member for being the first to go to college instead of pursuing the union or the force. I can't speak to how widespread this is, but being the first to make it to college is in itself sometimes a real achievement. It might be a wild thought for those of us who were raised thinking a Bachelor's is just part of the natural progression.

Conversely, in some minority groups it can be a great source of pride to be the first one in the family to go to college and get a bachelor's degree.

I don't think it's likely for the writer of the article to know that Jennifer McLoud was the first in her family to graduate college without McLoud revealing it herself. Automatically assuming it's racist seems like a stretch and unnecessary.

Agreed. Most of the time, perhaps all, I've heard this phrase, it's a source of pride.

We have to remember, the US was vastly and primarily agricultural and largely didn't attend secondary education in numbers till after WWII with the gi bill and subsequent popularization of secondary education.

The phrase is typically a way to show achievement in the face of slim odds, like from broken home, no tradition of emphasis on education, etc.

The only "racist" element, if it can be called that is the presumption the secondary education is seen as modern sophisticated and good, which I think it is, to practical extents, but can see where some may disagree.

One nit for your comment: In the US, "primary education" means grade school. "Secondary" means high school. "Post-secondary" is college. Through WWII, the majority of US children attended secondary education (high school) but not college.

True, it gets confusing, but tertiary to my mind means postgrad. But yeah, I meant college/university was rare before the post war period. And people thought and still consider it an achievement meriting praise.

Thank you. It still looks strange to me. Amorymeltzer’s addition in his answer here really did a lot IMHO.

> […] for those of us who were raised thinking a Bachelor's is just part of the natural progression

That’s probably my case. Thank you so much.

It's a common narrative, this is a phrase I've heard about many other first-in-the-family to be "educated". The intent is very likely honest, and simply meant to make Jennifer seem extra impressive.

I didn't take a racist interpretation immediately, though I see why it can be taken that way. I've always felt like this narrative is more classist. A family that has no bachelors degrees is another way to say "poor", and it's supposed to be surprising that someone from a poor family can get a university education and go on to do great things. To me it often seems like the implication is that socioeconomic status is an indicator of intelligence or academic capacity, or even the potential for it.

But my interpretion, and yours, are both cynical. The statement was almost certainly meant in good faith as a compliment, and on top of that, the sad truth is that the rates of people from non-white families who haven't gone to college breaking out and going to college are lower than the rates of middle class white kids going to college. It's not necessarily racist or classist to point that out, or to applaud someone for managing to break the trend; one might even argue the opposite.

Jennifer Mbutt-Mann is an Associate Professor in the Engineering & Mathematics department (http://www.bothell.washington.edu/engineering-math/faculty), and has a PhD.

It's often worthwhile to mention when someone is the first in their family to get a bachelor's because it's damn impressive; she not "only" did that but got a PhD! The full implication of the "wrong" statement is thus "..the first in her family to earn even a bachelor's degree, not to mention a doctorate or professorship."

I'm guessing you have that browser extension that converts the word "cloud" to "butt"?

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