what is math but a formalized set of nifty tricks.  braver has some challenges whose solutions tickled my fancy.  for example: diagonally slice the above parallelograms and rectangle and you end up with a triangle. turns out every triangle is just a quadrilateral cut in half, and thus the area will simply be half the area of it imaginary whole quadrilateral. the b \times h in \frac{1}{2}b \times h is sister to l \times b or l \times l of quadrilaterals. another, is this question: i was close to getting this but i was blocked in my mind by the focus on only the big square. turns out, the focus is the area of the big square, which can be seen as: the square of (a + b), which i quickly noticed or  (four times \frac{1}{2}(a \times b)) + c^2, which is just nice. equating these means:
a^2 + 2ab + b^2 = (4 \times \frac {1}{2} (a \times b)) + c^2 or,   a^2 + 2ab + b^2 = 2ab + c^2 \therefore a^2 + b^2 =  c^2 so nifty.  the equivalent of watching neymar at work.  this was based on his precalculus made difficult book. 

Nifty Triangle Geometry

question given the general quadratic equation: a x^2+b x+c=0
 derive the quadratic formula:  x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} solution \therefore \begin{aligned}& a x^2+b x+c=0 \\& x^2+\frac{b}{a} x+\frac{c}{a}=0\end{aligned} if complete squares are of the form: x^2+2 a x+a^2=(x+a)^2
 then, constant a^2=\left[\frac{2 a}{2}\right]^2 and complete squares become: x^2+2 a x+\left[\frac{2 a}{2}\right]^2=(x+a)^2 
for the second equation above, this gives: x^2+\frac{b}{a} x+\left[\frac{b}{ a}\times \frac1 2\right]^2+\frac{c}{a}=\left[\frac{b}{ a}\times \frac1 2\right]^2  x^2+\frac{b}{a} x+\frac{b^2}{4 a^2} + \frac{c}{a} =\frac{b^2}{4 a^2}  x^2+\frac{b}{a} x+\frac{b^2}{4 a^2} =\frac{b^2}{4 a^2} - \frac{c}{a} now the left hand side is a perfect square.
\left[x+\frac{b}{2a}\right]^2 = \frac{b^2}{4a^2}-\frac{c}{a} x+\frac{b}{2a} = \pm \sqrt{\frac{b^2}{4a^2}-\frac{c}{a}} x+\frac{b}{2a} = \pm \sqrt{\frac{b^2}{4a^2}-\frac{4ac}{4a^2}} x+\frac{b}{2a} = \pm \sqrt{\frac{b^2-4ac}{4a^2}} x+\frac{b}{2a} = \pm \frac{\sqrt{b^2-4ac}}{2a} x = \frac{-b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a} x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} voila.

Deriving the quadratic formula.

asking this question was based on unconscious ignorance of the shared aspect of a marriage with a real person you choose, instead implying a marriage with an amorphous amalgam of society’s implicitly communicated assumptions. with this implication, the hesitation and questioning was and still is reasonable. who would want to marry the many examples of terrible spouses you have heard of or seen. therefore, a better alternative is, “why marry this person you are choosing?”. perhaps there are benefits that bring one to a choice like this. an explicit discussion of a shared desirable future with a lover and friend. a desire for permanence in emotional connection with this person. economic and social benefits of being with them. what convinced me was the agency of the first benefit, since the desire for permanence of the relationship is implicit in my decision to be serious with the person in the first place, thus predating the choice of a marriage[1]and a focus on the economic and social benefits stems from a philosophy i am not interested in (i.e. using people). beyond that, a personally reasonable benefit is the elevation of your relationship. your friendship going beyond the less serious - yet important - relationship into spousehood is a personal, social, legal - and for those who believe, spiritual - testimony to the seriousness with which you intend to care for your spouse and your relationship.  if there is another way to proclaim this elevation that i am unaware of, i would love to know. therefore, it is not about the concept of a marriage, but the prospectus of a marriage to this person. the explicitness of this leads to a second question, what kind of marriage is worth undertaking such testimony? for me, only one is, me and my friend. me and my game, joke, drink, food, talk, serious, business, sex and pray buddy.[2] if i do not see the possibility of sharing myself in these ways with this person or wanting to know them in these many ways, there is no point to the entire enterprise. initially, i was asking the titular question from a purely epistemological perspective. i wanted to know why people did and do it, especially given the deluge of evidence against the choice. the good faith answer is that they do it because they believe in the person in front of them.[3] [1]: also, while marriage contains the promise of permanence, it is not an assurance of permanence, as seen by the normalcy of breakups and divorce. i am not certain how much this should weigh in consideration. [2]: it is why i propose asking out your good friend, not making your girlfriend your friend. not that that is not possible, but i have never fancied that.   [3]: i do not believe the primary purpose of marriage is begetting, hence the ignorance of children in this discussion. to me, they are a collaborative choice made after choosing a spouse.

Why marry?

Generating Molecular vdW Surfaces

it is crucial to be definite about tomorrow, to imagine a future worth living and wholly throw your self behind it. as i say this, i do not have the full idea or experience of what that is, but an attempt at this is likely better than having no definition. this thought was revived as i read peter thiel’s zero to one.  my receipt from this text is that specificity and optimism are learnable and important skills.   “if you treat the future as something definite, it makes sense to understand it in advance and to work to shape it. but if you expect an indefinite future ruled by randomness, you’ll give up on trying to master it.” — thiel, zero to one, 2014. attempt a definition of a great tomorrow and be adaptably committed to it, it will make tomorrow easier and more favorable. 

Define Tomorrow

work on problems worth it. you know them, you can feel it.  get the problems clear. build solutions that are tractable, scalable and bold. these require you to become persistent, okay with ambiguity and undistracted.  after succeeding, move on to new problems.  how not to work work on problems you don't care about, without change. work on important difficult problems without changing them to important yet tractable ones.  continuously excuse this hesitation to change.

2026

Nifty Triangle Geometry

Deriving the quadratic formula.

Why marry?

2025

Generating Molecular vdW Surfaces

Define Tomorrow

How To Work