Whatever Joey does blog

[jvang]

It's 2019!

Went out to practice driving for the first time today (with my mom in the car). First thoughts:

• The brakes and accelerator are very touchy, and it's very easy to cause the car to jerk back and forth.
• You can't see much out through the back window (we have a Honda CRV), so it would suck to not have a backup camera.
• You also can't see through a lot of the structural parts of the car, so there are a bunch of blind spots in your immediate vision.
• And the hood of the car makes it hard to park; I kept leaving 3-4 feet of space in the front of the parking spaces where I was practicing.

So not too bad, apart from the whole not-being-able-to-see-anything deal

[jvang]

I was going to post these in the cell phone astrophotography thread, but the cell phone ones were very bad. These were taken with a more legitimate camera:

[Uncwilly]

Very nice.

[CRGreathouse]

Originally Posted by jvang

• The brakes and accelerator are very touchy, and it's very easy to cause the car to jerk back and forth.

It gets easier with practice. Part of this is knowing how to angle your feet to get small gradual changes, part is just building endurance in small muscles so you don't get fatigue that makes your foot shake and the car surge.

Originally Posted by jvang

• You can't see much out through the back window (we have a Honda CRV), so it would suck to not have a backup camera.
• You also can't see through a lot of the structural parts of the car, so there are a bunch of blind spots in your immediate vision.
• And the hood of the car makes it hard to park; I kept leaving 3-4 feet of space in the front of the parking spaces where I was practicing.

So not too bad, apart from the whole not-being-able-to-see-anything deal

It's similar to your normal vision: you can't see much, and you have blind spots. But you learn to compensate by building an image of what is around you using the spots you can see, to the point that you hardly notice the blind spots.

[jvang]

My dad wanted me to post about my math observation; a long time ago I noticed that each successive square number is a successive odd number less than the next. So starting from 0, the first odd number is 1, which is the first square. Adding the next odd number, 3, to 1 gives 4, the second square. Add 5, get 9, add 7, get 16, etc. I’m not sure what kind of equation would describe this; it’d be trivial to code and easy to make a series describing it though. Is this a coincidental correlation or does it have some meaning?

[Uncwilly]

How long does this trend last?
Have you looked fro this in oeis?

[paulunderwood]

Originally Posted by jvang

My dad wanted me to post about my math observation; a long time ago I noticed that each successive square number is a successive odd number less than the next. So starting from 0, the first odd number is 1, which is the first square. Adding the next odd number, 3, to 1 gives 4, the second square. Add 5, get 9, add 7, get 16, etc. I’m not sure what kind of equation would describe this; it’d be trivial to code and easy to make a series describing it though. Is this a coincidental correlation or does it have some meaning?

Geometrically, think of adding one "layer" of ones to two adjacent sides of a square. I.e. for an nxn square add n to two adjacent sides and put one in the corner created. You have just added 2*n+1 to the nxn square and (n+1)*(n+1) = n*n+2*n+1.

[wblipp]

Originally Posted by jvang

I’m not sure what kind of equation would describe this

Have you learned Sigma notation yet? If so, you should be able to use it to write an expression for the sum of the first n odd numbers.

Have you learned inductive proofs yet? If so, you should be able to prove the relationship holds.

[Nick]

Originally Posted by jvang

Is this a coincidental correlation or does it have some meaning?

Good observation!

Here's a picture to see what's going on:


If we add this to the observation you made earlier we have 2 sequences:
\[
\begin{eqnarray*}
1+2+3+\ldots +n & = & \frac{1}{2}n(n+1) \\
\underbrace{1+3+5+\ldots +(2n-1)}_n & = & n^2
\end{eqnarray*}
\]
In both sequences, the difference between terms next to each other remains constant
(the numbers go up by 1 each time in the first sequence and by 2 each time in the second one).

Sequences with this property are known as arithmetic progressions.
There is a formula for the sum of all the terms in any arithmetic progession, which is useful to know:
add the first and last terms together, multiply by the number of terms and divide by 2