#### December 4, 2017

We are trying the Hour of Code today. Updates to follow.

**Citing this post**: **Urbano**, L., 2017. Hour of Code, Retrieved December 11th, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in ProgrammingNo Comments » - Tags: programming

#### June 29, 2017

3d model of the Hawaiian island chain, rendered in OpenSCAD.

After a lot of hours of experimentation I’ve finally settled on a workable method for generating large-scale 3d terrain.

**Data** from the NGDC’s Grid Extraction tool. The ETOPO1 (bedrock) option gives topography and bathymetry. You can select a rectangle from a map, but it can’t be too big and, which is quite annoying, you can’t cross the antimeridian.

The ETOPO1 data is downloaded as a GeoTIFF, which can be easily converted to a png (I use ImageMagick convert).

The Hawaiian data with the downloaded grayscale.

**Adjusting the color scale**. One interesting property of the data is that it uses a grayscale to represent the elevations that tops out at white at sea-level, then switches to black and starts from there for land (see the above image). While this makes it easy to see the land in the image, it needs to be adjusted to get a good heightmap for the 3d model. So I wrote a python script that uses matplotlib to read in the png image as an array and then I modify the values. I use it to output two images: one of the topography and one of just land and water that I’ll use as a mask later on.

Hawaiian Islands with adjusted topography and ocean-land.

The images I export using matplotlib as grayscale png’s, which can be opened in OpenSCAD using the surface command, and then saved as an stl file. Bigger image files are take longer. A 1000×1000 image will take a few minutes on my computer to save, however the stl file can be imported into 3d software to do with as you will.

Note: H.G. Deitz has a good summary of free tools for Converting Images Into OpenSCAD Models

**Citing this post**: **Urbano**, L., 2017. Generating 3d Terrain, Retrieved December 11th, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Makerspace, TechShopNo Comments » - Tags: 3d models, programming, python, TechShop

#### March 29, 2017

Students playing Go.

I recently discovered that, although they may look it, Go boards are not necessarily square. They’re slightly longer in one dimension so that the board looks more square to the players on both sides.

A student asked me to make one for him–he’d ordered a set recently and didn’t like the board it came with–so, I wrote a small python program to generate the Go grid, then lasered it onto a nice piece of sanded plywood.

It worked out quite well. Apparently the plywood makes just the right “thunk” sound when you put down the pieces.

Go board in use.

The script to generate the grid.

*go_board_2.py*

from visual import *
from svgInator_3 import *
length = 424.2 #mm
width = 454.5 #mm
nLines = 19
dx = length/(nLines-1)
dy = width/(nLines-1)
print "Lenght = ", length
print "dx = ", dx
f = svgInator("go_board.svg")
lineStyle = {"stroke": "#000", "stroke-width": "2pt",}
#lines
for i in range(nLines):
x = i * dx
y = i * dy
#vertical
f.line(pos=[vector(x,0), vector(x,width)], style=lineStyle)
#horizontal
f.line(pos=[vector(0,y), vector(length,y)], style=lineStyle)
#circles
grid_pos = [(3,3), (3,9), (3,15),
(9,3), (9,9), (9,15),
(15,3), (15,9), (15,15)]
for i in grid_pos:
(x, y) = (i[0]*dx, i[1]*dy)
f.circle(pos=vector(x,y), radius=2.0,
style={"stroke": "#000", "fill":"#000"})
#bounding box
f.rect(dim=vector(length,width), style=lineStyle)
f.close()

Now I just have to learn to play.

**Citing this post**: **Urbano**, L., 2017. Go Board, Retrieved December 11th, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Makerspace, TechShopNo Comments » - Tags: board games, geometry, makerspace, programming, TechShop

#### February 23, 2017

Temperature change over time (in thousands of years). As the Earth warms from 3K to equilibrium.

If the Earth behaved as a perfect black body and absorbed all incoming solar radiation (and radiated with 100% emissivity) the we calculated that the average surface temperature would be about 7 degrees Celsius above freezing (279 K). Keeping with this simplification we can think about how the Earth’s temperature could change with time if it was not at equilibrium.

If the Earth started off at the universe’s background temperature of about 3K, how long would it take to get up to the equilibrium temperature?

Using the same equations for incoming solar radiation (E_{in}) and energy radiated from the Earth (E_{out}):

Symbols and constants are defined here except:

At equilibrium the energy in is equal to the energy out, but if the temperature is 3K instead of 279K the outgoing radiation is going to be a lot less than at equilibrium. This means that there will be more incoming energy than outgoing energy and that energy imbalance will raise the temperature of the Earth. The energy imbalance (ΔE) would be:

All these energies are in Watts, which as we’ll recall are equivalent to Joules/second. In order to change the temperature of the Earth, we’ll need to know the **specific heat capacity** (c_{E}) of the planet (how much heat is required to raise the temperature by one Kelvin per unit mass) and the mass of the planet. We’ll approximate the entire planet’s heat capacity with that of one of the most common rocks, granite. The mass of the Earth (m_{E}) we can get from NASA:

- c
_{E = 800 J/kg/K}
- m
_{E} = 5.9723×10^{24}kg

So looking at the units we can figure out the the change in temperature (ΔT) is:

Where Δt is the time step we’re considering.

Now we can write a little program to model the change in temperature over time:

*EnergyBalance.py*

from visual import *
from visual.graph import *
I = 1367.
r_E = 6.371E6
c_E = 800.
m_E = 5.9723E24
sigma = 5.67E-8
T = 3 # initial temperature
yr = 60*60*24*365.25
dt = yr * 100
end_time = yr * 1000000
nsteps = int(end_time/dt)
Tgraph = gcurve()
for i in range(nsteps):
t = i*dt
E_in = I * pi * r_E**2
E_out = sigma * (T**4) * 4 * pi * r_E**2
dE = E_in - E_out
dT = dE * dt / (c_E * m_E)
T += dT
Tgraph.plot(pos=(t/yr/1000,T))
if i%10 == 0:
print t/yr, T
rate(60)

The results of this simulation are shown at the top of this post.

What if we changed the initial temperature from really cold to really hot? When the Earth formed from the accretionary disk of the solar nebula the surface was initially molten. Let’s assume the temperature was that of molten granite (about 1500K).

Cooling if the Earth started off molten (1500K). Note that this simulation only runs for 250,000 years, while the warming simulation (top of page) runs for 1,000,000 years.

**Citing this post**: **Urbano**, L., 2017. Modeling Earth's Energy Balance (Zero-D) (Transient), Retrieved December 11th, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Natural World, ProgrammingNo Comments » - Tags: modeling, numerical methods, physics, programming

#### February 22, 2017

For conservation of energy, the short-wave solar energy absorbed by the Earth equals the long-wave outgoing radiation.

Energy and matter can’t just disappear. Energy can change from one form to another. As a thrown ball moves upwards, its kinetic energy of motion is converted to potential energy due to gravity. So we can better understand systems by studying how energy (and matter) are conserved.

## Energy Balance for the Earth

Let’s start by considering the Earth as a simple system, a sphere that takes energy in from the Sun and radiates energy off into space.

### Incoming Energy

At the Earth’s distance from the Sun, the incoming radiation, called insolation, is 1367 W/m^{2}. The total energy (wattage) that hits the Earth (E_{in}) is the insolation (I) times the area the solar radiation hits, which is the area a cross section of the Earth (A_{cx}).

Given the Earth’s radius (r_{E}) and the area of a circle, this becomes:

### Outgoing Energy

The energy radiated from the Earth is can be calculated if we assume that the Earth is a perfect black body–a perfect absorber and radiatior of Energy (we’ve already been making this assumption with the incoming energy calculation). In this case the energy radiated from the planet (E_{out}) is proportional to the fourth power of the temperature (T) and the surface area that is radiated, which in this case is the total surface area of the Earth (A_{surface}):

The proportionality constant (σ) is: σ = 5.67 x 10^{-8} W m^{-2} K^{-4}

Note that since σ has units of Kelvin then your temperature needs to be in Kelvin as well.

Putting in the area of a sphere we get:

### Balancing Energy

Now, if the energy in balances with the energy out we are at equilibrium. So we put the equations together:

cancelling terms on both sides of the equation gives:

and solving for the temperature produces:

Plugging in the numbers gives an equilibrium temperature for the Earth as:

T = 278.6 K

Since the freezing point of water is 273K, this temperature is a bit cold (and we haven’t even considered the fact that the Earth reflects about 30% of the incoming solar radiation back into space). But that’s the topic of another post.

**Citing this post**: **Urbano**, L., 2017. Modeling Earth's Energy Balance (Zero-D) (Equilibrium), Retrieved December 11th, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Natural World, ProgrammingNo Comments » - Tags: computer models, modeling, numerical methods, programming

#### January 10, 2017

Paths of a projectile.

I had my Numerical Methods student calculate the angle that would give a ballistic projectile its maximum range, then I had them write a program that did the the same by just trying a bunch of different angles. The diagram above is what they came up with.

It made an interesting pattern that I converted into a face-plate cover for a light switch that I made using the laser at the TechShop.

Face plate cover.

**Citing this post**: **Urbano**, L., 2017. Projectile Paths, Retrieved December 11th, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Physics, ProgrammingNo Comments » - Tags: ballistics, math, numerical methods, programming, TechShop

#### November 29, 2016

This TedEd video explains a few common sorting methods used in computer science. Sorting can be a challenging computational problem because of the enormous number of comparisons between items that can be involved, so computer scientists has spent a lot of time looking into it.

The video below shows the Quick Sort method using Hungarian folk dance.

**Citing this post**: **Urbano**, L., 2016. Sorting algorithms (with Hungarian Dance), Retrieved December 11th, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Programming, videoNo Comments » - Tags: programming

#### November 3, 2016

Previously, I showed how to solve a simple problem of motion at a constant velocity analytically and numerically. Because of the nature of the problem both solutions gave the same result. Now we’ll try a constant acceleration problem which should highlight some of the key differences between the two approaches, particularly the tradeoffs you must make when using numerical approaches.

**The Problem**

- A ball starts at the origin and moves horizontally with an acceleration of 0.2 m/s
^{2}. Print out a table of the ball’s position (in x) with time (every second) for the first 20 seconds.

**Analytical Solution**

We know that acceleration (*a*) is the change in velocity with time (*t*):

so if we integrate acceleration we can find the velocity. Then, as we saw before, velocity (*v*) is the change in position with time:

which can be integrated to find the position (x) as a function of time.

So, to summarize, to find position as a function of time given only an acceleration, we need to integrate twice: first to get velocity then to get x.

For this problem where the acceleration is a constant 0.2 m/s^{2} we start with acceleration:

which integrates to give the *general solution*,

To find the constant of integration we refer to the original question which does not say anything about velocity, so we assume that the initial velocity was 0: i.e.:

at *t = 0* we have *v = 0*;

which we can substitute into the velocity equation to find that, for this problem, c is zero:

making the *specific* velocity equation:

we replace *v* with *dx/dt* and integrate:

This constant of integration can be found since we know that the ball starts at the origin so

at *t = 0* we have *x = 0*, so;

Therefore our final equation for x is:

### Summarizing the Analytical

**To summarize** the analytical solution:

These are all a function of time so it might be more proper to write them as:

Velocity and acceleration represent rates of change which so we could also write these equations as:

or we could even write acceleration as the second differential of the position:

or, if we preferred, we could even write it in prime notation for the differentials:

## The Numerical Solution

##

As we saw before we can determine the position of a moving object if we know its old position (*x*_{old}) and how much that position has changed (*dx*).

where the change in position is determined from the fact that velocity (*v*) is the change in position with time (*dx/dt*):

which rearranges to:

So to find the new position of an object across a timestep we need two equations:

In this problem we don’t yet have the velocity because it changes with time, but we could use the exact same logic to find velocity since acceleration (*a*) is the change in velocity with time (*dv/dt*):

which rearranges to:

and knowing the change in velocity (dv) we can find the velocity using:

Therefore, we have four equations to find the position of an accelerating object (note that in the third equation I’ve replaced *v * with *v*_{new} which is calculated in the second equation):

These we can plug into a python program just so:

*motion-01-both.py*

from visual import *
# Initialize
x = 0.0
v = 0.0
a = 0.2
dt = 1.0
# Time loop
for t in arange(dt, 20+dt, dt):
# Analytical solution
x_a = 0.1 * t**2
# Numerical solution
dv = a * dt
v = v + dv
dx = v * dt
x = x + dx
# Output
print t, x_a, x

which give output of:

>>>
1.0 0.1 0.2
2.0 0.4 0.6
3.0 0.9 1.2
4.0 1.6 2.0
5.0 2.5 3.0
6.0 3.6 4.2
7.0 4.9 5.6
8.0 6.4 7.2
9.0 8.1 9.0
10.0 10.0 11.0
11.0 12.1 13.2
12.0 14.4 15.6
13.0 16.9 18.2
14.0 19.6 21.0
15.0 22.5 24.0
16.0 25.6 27.2
17.0 28.9 30.6
18.0 32.4 34.2
19.0 36.1 38.0
20.0 40.0 42.0

Here, unlike the case with constant velocity, the two methods give slightly different results. The analytical solution is the correct one, so we’ll use it for reference. The numerical solution is off because it does not fully account for the continuous nature of the acceleration: we update the velocity ever timestep (every 1 second), so the velocity changes in chunks.

To get a better result we can reduce the timestep. Using *dt = 0.1* gives final results of:

18.8 35.344 35.532
18.9 35.721 35.91
19.0 36.1 36.29
19.1 36.481 36.672
19.2 36.864 37.056
19.3 37.249 37.442
19.4 37.636 37.83
19.5 38.025 38.22
19.6 38.416 38.612
19.7 38.809 39.006
19.8 39.204 39.402
19.9 39.601 39.8
20.0 40.0 40.2

which is much closer, but requires a bit more runtime on the computer. And this is the **key tradeoff** with numerical solutions: *greater accuracy requires smaller timesteps which results in longer runtimes on the computer*.

### Post Script

To generate a graph of the data use the code:

from visual import *
from visual.graph import *
# Initialize
x = 0.0
v = 0.0
a = 0.2
dt = 1.0
analyticCurve = gcurve(color=color.red)
numericCurve = gcurve(color=color.yellow)
# Time loop
for t in arange(dt, 20+dt, dt):
# Analytical solution
x_a = 0.1 * t**2
# Numerical solution
dv = a * dt
v = v + dv
dx = v * dt
x = x + dx
# Output
print t, x_a, x
analyticCurve.plot(pos=(t, x_a))
numericCurve.plot(pos=(t,x))

which gives:

Comparison of numerical and analytical solutions using a timestep (dt) of 1.0 seconds.

**Citing this post**: **Urbano**, L., 2016. Numerical and Analytical Solutions 2: Constant Acceleration, Retrieved December 11th, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Calculus, Mathematics, ProgrammingNo Comments » - Tags: motion, numerical methods, physics, physics and calculus, programming