I had been working on a problem that he posted about regarding a bead sliding freely on a hoop that is spinning about an axis in its plane that goes through the center. That’s a pretty typical Lagrangian Dynamics-type problem and I wondered what would happen if the hoop wasn’t driven to constantly go with a particular angular frequency but rather was set spinning on that axis with that initial angular frequency but after that it would respond dynamically (speed up or slow down its rate of spinning) according to the physics involved. Here’s what that looks like:

But then when Rhett asked about letting the hoop be even more free to rotate, in other words not just around that one axis, I realized I’d have to calculate things a different way.

That led me to remind myself how to do Lagrangian Dynamics with rigid bodies, mostly dealing with things like the Inertia Tensor and body axes. This post is to make it so I don’t have to start from scratch with that again. Note that this post lays out the broad strokes of what I’m doing here.

For those of you hoping I’ll derive the Euler rotation equations, I should let you know that I don’t explicitly do that here. I get close, but I really move in a direction where I can solve the equations of motion and make some fun vids of the motion.

Here’s the steps involved:

- Determine the generalized coordinates
- Determine the kinetic and potential energies
- Use the Principal Axes (actually you don’t
**have**to do that) - Do all the usual Lagrangian stuff

If you need to describe the orientation of a rigid body, how would you do it? I’ve done this with my students by asking them to all grab their chairs and figure out a systematic way to orient them the way mine is (which I do when they have their eyes closed). At first students think you just need the polar angles and because they figure that all the r’s are fixed and so you don’t need that third polar coordinate. But then I show them two situations with my chair that share the polar angles but look different. Ultimately my students usually land on a version of the Euler angles, which are really just the polar angles with a final twist of the body around the original vertical axis. Most texts call that last angle .

Well, if the Euler angles are the information that you need to describe the orientation of a rigid body, that means they’re just the perfect generalized coordinates for a Lagrangian Dynamics approach. All we have to be able to do is describe the potential and kinetic energy of the object as functions of the Euler angles and we’re good to go!

Early on physics students learn that if they’re dealing with rotating bodies they can replace with . Essentially what happens is that you realize that all parts of the system share an angular speed () even if different parts are traveling with different linear speeds (v). Since they all share the same thing, you can pull that out of any sums or integrals you’re doing to add up all the kinetic energy and whatever’s left you just lump into something we call the moment of inertia ().

The problem is that for oddly shaped things that simple approach doesn’t quite work. Sure all the pieces and parts still share an angular velocity (now we’re talking about the vector since the axis of rotation is something we need to know), but it’s not as simple as . Probably the easiest way to see that is that the angular momentum is not necessarily parallel to the angular velocity (STOP: read that again, as it’s one of the single most important aspects of rigid body motion and it sounds really weird to most people). Instead there’s a relationship between the angular momentum vector and the angular velocity vector that’s more complicated. If you know the angular velocity you can find the angular momentum with a 3D rotation, and that’s done computationally with a 3×3 tensor which we call the inertia tensor. Then you have . Essentially a 3×3 tensor dotted into a vector gives you a new vector that (often) points in a new direction (hence them not being parallel anymore).

Really the inertia tensor has just what we need. Just like we can do for kinetic energy normally (try it!), we can do or:

Aha! So now if we know how both the inertia tensor and the angular velocity vector can be described by the Euler angles (and their time derivatives) we’re in business!

The inertia tensor is certainly calculate-able if you know the orientation of the object (which you do if you know the Euler angles). Essentially you can figure out where all the particles of the body are and then do all the integrals necessary for the inertia tensor (remember, the tensor is just made up of what’s left after you factor out the common terms that all the particles share – namely the angular velocity – so it’s just a collection of sums/integrals involving the locations of the particles). What really sucks, though, is that at every point in time there is a new orientation and hence a whole new set of integrals to do. Ugh. Don’t worry though, that gets a lot easier in the next section!

The angular velocity is certainly a function of the time derivatives of the Euler angles since as they change they rotate the body, and rotation is all that angular velocity is used to describe. Here’s the cool part: the angular velocity () is equal to the sum of three vectors, one for each Euler angle. Each Euler angle contributes it’s time derivative (which is exactly the right units for angular velocity) in the direction of the instantaneous axis of rotation that Euler angle rotates around. I know, that was complicated, but it’s relatively straightforward to work out:

First consider the easiest one: . When you think about it that polar vector, all it does is rotate everything around the vertical-, or z-, axis. Therefore ‘s contribution to is .

Next consider : When it changes, it tilts the object further from the vertical. But it does so in the plane determined by . The instantaneous axis of rotation is the location of the new y-axis after the rotation. So that means that the contribution to is

Finally we have . It rotates the body around its original vertical axis. That means its instantaneous axis of rotation is wherever its vertical axis has gotten to after the polar angles have done their thing. So ‘s contribution is .

All together then we have these for the components of the angular velocity vector:

Ok, we have the angular velocity all set to go, but we saw how the inertia tensor, while definitely a function of the Euler angles, might have to be calculated over and over again. What if we only had to do it once!

The trick is to decide to think about how things feel in the frame of the rigid body. In what we call the “body-frame” it sits there while the world moves around it. Specifically the angular velocity vector is all over the place. But, and this is the important part, the inertia tensor is fixed! You just have to calculate it once in that orientation. So the problem becomes determining what the angular velocity vector (which we calculated above) looks like in that body-frame.

But luckily the Euler angles were built to transform any vector into a new rotated state. So we just need to apply the Euler rules to the angular velocity vector above. Except that we’re trying to describe how things look to the body frame, so what we really need is the reverse of the Euler rules, namely:

- negative rotation of around the z-axis, followed by. . .
- a negative rotation of around the y-zaxis, followed by . . .
- a negative rotation of around the z-axis

I know, that seems weird, but, trust me, it does the trick. All of those rotations are pretty straightforward for something like Mathematica to do so we’re in business: Calculate the inertia tensor once, figure out the lab-based angular velocity vector, and then do the steps above to find the body-frame version of the angular velocity vector. Then you calculate the kinetic energy.

Here’s what that looks like in Mathematica:

After the “KE=…” line is the usual construction of how I do Lagrangian Mechanics. The “Method->…” stuff was suggested by Mathematica after the first solve seemed to fail. That Method fix did the trick.

You’ll note that my inertia tensor (“Ithing”) is really simple. There are no off-diagonal terms in the matrix because I wanted to investigate the same sort of thing I did in this post about flipping handles in space. The section below called “Principal Axes frame” is how I teach my students to find the magical orientiation of an object where its inertia tensor has that simple setup. Note that it’s not necessary for the work I’ve laid out here.

Note that while this makes it pretty do-able, you’re still going to want a Computer Algebra System to help. Here’s just the Euler-Lagrange equation (all of this equals zero) for a generic inertia tensor:

Ok, so now I had the tools to answer Rhett’s question. I’ve got an interesting system made up of a hoop and a bead. What I decided to do was to fix for the hoop at zero because otherwise I would have to keep track of the bead’s sliding separate from the rotation of the hoop around it’s (original) vertical axis. The orientation of the hoop when all the Euler angles are zero is horizontal. So I first set to to stand it up and then I set to some initial spin rate.

For the bead I just determine it’s contribution to the inertia tensor of the system. Luckily that’s just a simple function of (remember I fix that at zero for the hoop). So I basically do the same thing as before. Here’s the Mathematica:

Really I have to carefully do the hoop and the bead separately when doing the calculations, but otherwise it’s similar to what I did above.

Here’s the results!

You can see how the hoop starts to tilt over. It has to to maintain a constant angular momentum (since there’s no torque in the problem if you don’t fix the axis of the hoop).

I noted above that you don’t have to use the principal axes frame but I wanted to put this in for my future self:

When I teach this I point out how difficult things are when the angular moment is not parallel to the angular velocity. I ask them if they’d like it if we could make that disappear. Usually they say yes.

I tell them that there are some magic scenarios when the two are parallel. Essentially there are three axes that if you rotate around them you get an angular momentum in that same direction. There are a few ways to figure them out, but most are either difficult to understand mathematically (eigenvectors of the inertia tensor) or would take forever (just spin the object along all possible axes and look to see when the angular momentum is along that same axis). But there’s one way I’ve found that my students can do. I have to explain how a diagonalized inertia tensor does the trick. I do that by saying that we’ll just keep orienting the object until we find that all three of the magic axes are along the normal lab axes (x-, y-, and z-axes). In that orientation, you get what we need as long as there are no non-zero off-diagonal components of the inertia tensor.

So I provide my students with a Mathematica document that lets them freely adjust the Euler angles for an object and it constantly recalculates the inertia tensor for that orientation (see the last section above to understand why the tensor changes with orientation). What’s cool is that they can see how certain adjustments to the Euler angles tends to reduce at least one of the 3 off-diagonal components and they usually quickly find a systematic way to get them all to be vanishingly small. When they do that, they’re staring at the orientation that puts the magic axes along the lab axes!

I’d love to hear what you think. Here are some starters for you:

- I love this! What I especially am surprised by is . . .
- This is dumb. I especially hated . . .
- So whatever Rhett says you just do? That sounds interesting . . .
- Once again no python, loser
- I thought you only had a chromebook at home. How are you doing all this Mathematica?
- Whoa, it looks like you do the rotation first in your Mathematica code. Didn’t you say it comes last?
- Wait, I don’t have to diagonalize the inertia tensor before doing these kinds of simulations? That’s so cool!
- Do you provide tools to help your students unbolt their chairs from the desks?
- I want to hear more about how students can systematically find the principal axes by just adjusting the Euler angles!
- Why don’t you have to do the normal accelerated frame adjustments for your kinetic energy? (it’s quite late as I type this and I realize I don’t have a good answer beyond “it just works”)

Here’s the basic gist of what I did:

- Get the lat/long of my house to have a starting point
- Do some geo-based math to find points on a regular polygon that includes my house. I set the polygon perimeter length to my target length
- Ask openrouteservice to find directions to all of the points in order (beginning and ending at my house)
- Adjust the total polygon perimeter until the actual path is my target distance
- Construct a google maps URL so that I can navigate while riding

This one is easy. Just open google maps and right click anywhere to get the latitude and longitude

After doing that here’s what’s in your clipboard: 44.9240104020827, -93.09348080561426

So just paste that somewhere in your python code as a list and you’re good to go.

I’m pretty sure I could have used some package to do this, but I figured it wouldn’t be that hard to just code up a sloppy version, especially since I knew I’d have to make adjustments to the polygon positions so that the actual travel distance would be what I wanted (see step 4).

If you have the lat/long of a point and want to find a point some known distance away in some particular direction (say degrees counterclockwise from east), you need to realize that while moving north/south follows a great circle (with a radius equal to the earth’s), moving east/west is not moving along a great circle (which is why you have to turn slightly north when driving straight east). Instead you’re moving on a circle whose radius is the earth’s radius multiplied by the cosine (ugh) of the latitude. If you were thinking it should have been sine (like my first thought) it’s likely because you’re used to using polar angles in mathematics (ie, the zero of latitude is the equator, not the north pole). Also, as you’ll note looking at the lat/long I pasted in above, since I’m roughly at 45 degrees latitude, it doesn’t really matter.

So here’s how I coded it:

```
earthRad=6378.1
def newPoint(start, angle, distance):
[long,lat]=start
horizScale=earthRad*math.cos(lat*math.pi/180)
newLong= long+distance*math.cos(angle)/horizScale*180/math.pi
newLat=lat+distance*math.sin(angle)/earthRad*180/math.pi
return [newLong,newLat]
```

Ok, so now I just need to make a bunch of points on a regular polygon:

```
def polygon(sides,start,totalDistance,initialAngle=0):
theList=[start]
curLoc=start
for x in range(sides):
curLoc=newPoint(curLoc,x/sides*2*math.pi+initialAngle,totalDistance/sides)
theList.append(curLoc)
return theList
```

You might wonder about that “initialAngle” part. I found that being able to tilt the initial polygon helped because I have a big river just east of me that the mapping software doesn’t like (only if you try to do bicycle navigating right into the river).

Here’s the code I used:

```
import openrouteservice as ors
from openrouteservice import convert
import folium
import math
token="GET YOURS FROM OPENROUTESERVICE"
client=ors.Client(key=token)
home=[PASTE THIS IN FROM GOOGLE MAPS]
directions=client.directions(profile='cycling-regular', coordinates=polygon(7,home[::-1],60,3*math.pi/4))
directions['routes'][0]['summary']['distance']
```

You can get your own token by signing up (for free!) at https://openrouteservice.org/ It has some limitations regarding how often you can use it, but they’re very generous, at least for this kind of thing.

The last line is what you have to do to get the total distance for the journey. You can see that the result of the “client.directions” command is a highly structured object (it took me a while to figure out how to extract that total distance). You get the map I generated above with this:

```
map=folium.Map(width=500, height=500,location=home)
folium.GeoJson(convert.decode_polyline(directions['routes'][0]['geometry'])).add_to(map)
map
```

Now you see why I had to import folium up above.

You’ll note in the code above that I submitted points on a polygon that would have a total perimeter of 60 kilometers, but it returned a path that’s just over 80 km, or 50 miles (my original goal). The reason is mostly due to the taxi-cab geometry I talk about in my 2nd approximation in this post. In other words, there’s no way there are dead straight roads among all the polygon vertices, so you’re going to travel further.

While I like the programmatic capabilities of openrouteservice, I *love* google maps as a navigation aid while riding. So I wanted to figure out how to get my newly found route into google maps. Solution: use their url api! On that page you can learn how to create URLs with starting points, ending points, way points (points along the journey), and type of directions (I wanted bicycle directions). Here’s how I built that:

```
pgon=polygon(7,home[::-1],60,1*math.pi/4)
baseUrl="https://www.google.com/maps/dir/?api=1&"
start="origin="+str(home[0])+","+str(home[1])
travelmode="travelmode=bicycling"
waypoints="waypoints="+"|".join([str(x[1])+","+str(x[0]) for x in pgon])
destination="destination="+str(home[0])+","+str(home[1])
baseUrl+start+"&"+travelmode+"&"+waypoints+"&"+destination
```

That gives you a URL that produces this:

And I’m finally ready to ride!

It was a beautiful late October day and I really did have a blast. The mapping worked really well but there are some issues I had to deal with.

If you look at the image above, you’ll probably note the little spur on the eastern edge of the route. Essentially google’s mapping algorithm (really both algorithms if you go back and look at the first image on this post) didn’t know of a way to get to and from that particular waypoint without a little bit of doubling back. I didn’t want to do that, so I just figured I could delete that waypoint (google makes that pretty easy to do). However, if you delete it before you begin, google will try to find a faster way from the prior waypoint to the next one, so I just waited until I got close before deleting it. That worked well.

The other waypoints weren’t so far off a normal path, but I did do a lot of weird extra blocks to get to the exact point I asked for. I probably could have saved a few miles not doing that, but that wasn’t really the point.

Overall I think it worked out pretty well. I’m excited to do it again. I’ll probably just tilt the 7-sided polygon over just a little further to get a completely different ride.

Got any feedback for me? Here are some starters for you:

- I love this! Can you do it for . . .?
- I hate this! It’s obvious you could do this in a much easier fashion by . . .
- What do you mean you have to turn a little north to drive straight east?
- I think we should redo latitude so that the zero is at the north pole
- Why python, I thought you loved Mathematica?
- Why are the google and openrouteservice directions slightly different?
- Why only 50 miles?
- How long did it take you? (8 hours but that was with a really long lunch and a few other stops to read and watch rugby)
- How did you find rugby to watch?
- What’s wrong with using cosine (you said “ugh” after doing so)?

I asked, on Facebook, whether filling it with paint would essentially be painting it on the inside and I had a suspicion that my good friend Art Guetter (Professor of Mathematics at my institution) would help me learn something. I was right, and he’s been kind enough to type up his thoughts for this post. Here he shows that it’s pretty tough to use a finite amount of paint to paint anything infinitely long.

Gabriel’s horn is a solid created by rotating the graph of , defined on the interval , around the -axis. Two “paradoxes” arise. In the first, the horn has a finite volume, despite being created by rotating a region with infinite area around an axis. The second is that the horn has finite volume but infinite surface area, leading to the apparent paradox that the horn could be filled with a finite volume of paint, yet the paint would not be sufficient to coat (that is, paint) the surface. The resolution of this “painter’s paradox” is that the thickness of the paint would need to decrease to 0 in the limit as tends to infinity. The assumption being made here is that painting requires a uniform thickness of paint. Note that I can paint the entire plane if I am allowed to decrease the thickness of paint as I move far from the origin.

So could I paint an infinite solid of revolution (to a uniform depth ) if the surface area were finite? As a first example, replace from Gabriel’s horn with a piecewise constant function when for , and the constants to be determined later. The surface will consist of an infinite collection of right circular cylinders, and each cylinder will have surface area . If the are chosen so that the sum , can I paint the surface with a finite amount of paint? The answer appears to be “yes”, but this involves the assumption that I roll each cylinder open, so that the amount of paint used is simply the surface area multiplied by the thickness of the paint, say . (Each cylinder can be rolled open without issue because they have thickness 0.)

What about painting if the cylinders aren’t rolled out? I will assume that painting to a thickness means that the depth of paint at any point is measured along the normal, in the outward direction. The amount of paint needed to paint one of the cylinders is then given by , where is the surface area of the cylinder and is the mean curvature of the cylinder. Summing this over will lead to an infinite volume of paint, no matter how fast the tend to 0.

A more general theorem has that the volume of a surface that has been thickened by an amount in the direction of the normal to the surface (assuming that is small enough that there is no self-intersection) is given by

where is the total surface area, is the surface element, is the mean curvature of the surface, and is the Gaussian curvature. (These are constant for the cylinder, with values and .) The amount of paint needed to paint a surface to uniform thickness depends on the curvature of the surface.

For a surface created by revolving the curve around the -axis, the values of and depend only on and are given by (in general and then for )

]]>Last weekend I went hammock camping by towing all my gear behind my bike. I loved it and now I’m interested in finding other adventures that won’t tax me too much. I really think that, for now, 30 miles in one day towing the trailer is a good limit for me. It leaves me enough energy to make camp and I’m able to relax and enjoy myself.

My first thought was to just look at a map with a 30-mile radius circle centered on my house. I figured if I could find any campgrounds in that circle I’d be good to go.

The problem with this approach is that there aren’t any roads or paths that go straight from my house to the edge of this circle. Any way I’d ride out to that edge would be more than 30 miles of riding.

I’ve noticed that I spend most of my time riding in the Cardinal directions (North, South, East, and West). What If I could figure out how far I could ride only going in those directions?

I realized that I could do a polar plot around my house if I could find a relationship between r (the radius) and theta (the angle with respect to East). It took a little head scratching, but here’s what I came up with:

or

Actually that only works in the first quadrant. For all the quadrants you want the absolute value of both the cosine and sine term:

Weird and surprising, right? I hope it was for at least a few of you. It surprised me! But of course, after some thought it makes some sense. As you give up height you get exactly that much more width. So a straight line with a slope of 45 degrees it is!

So why doesn’t the blue square touch the red circle, you might be asking? Well, I’m not really sure. I got the locations of those 4 corners from Google Maps using the “Measure Distance” feature, whereas the red circle comes from the Circle command in python’s folium library. It’s weird that they’re so far off from each other, isn’t it?

The obvious problem with this solution is that not all roads/paths run along the Cardinal directions. I think that’s likely even more true for bike paths.

I realized that Mathematica has the command TravelDirections that lets you put in two locations and a TravelMethod to get decent directions. I used “Biking” for the TravelMethod for all of this work.

I had Mathematica get directions for 36 evenly spaced locations on that red circle above. Then I took a look at the actual travel distance and they were all well over 30 miles. So then, for all 36 of those, I crept in towards my house by 1 kilometer until I got a path that was miles away. Then I just plotted them all to get a decent sense of my options:

So I think this is my best approximation yet. It gives me a real good sense of how far I can get comfortably in one day. I haven’t really investigated the directions, but certainly I can see the exact path I took this past weekend in there (it’s the one that heads just a little west of south)

Ok my mind was just blown with something called isochrones in the OpenRouteService available to python. Here’s my colab. And here’s the amazing result:

So what do you think? Here are some starters for you:

- This is cool! Have you thought of doing this . . .?
- This is dumb. Taxi drivers figured this out years ago
- Why do you only sometimes capitalize the carDinAl directions?
- Could you please share the colab doc you used for the folium maps?
- Could you please share your Mathematica code (saved at work and I’m typing this at home for the moment)?
- I live ____ and can ride ___ miles in a day. Could you do this for me?
- I know why the blue and red don’t touch, it’s because . . .
- That blue square didn’t surprise me at all. You’re dumb.
- That blue square totally surprised me! I learned something today!
- You’re going to tease us about your bike and not bother showing a pic?

- Allow my students to take a picture of their work and share it with the class
- Certainly my computer should be able to display it, but with thumbnails for all the images
- Bonus if all the images are on everyone’s device

This post is about my attempt to make just such an app in Google Apps Script. It includes detailed instructions for how you can make your own copy. If you want to see it in action, see this vid.

Before jumping in, I want to say thanks to some great twitter friends who had thoughts about other ways to do this. I haven’t tried them all, but I think they all could work really well:

Below I’ll explain a little more about how it works, but one important feature is the use of webSockets. To do that I use a free account on pusher.com. It limits all my apps to 100 total simultaneous connections. For the purposes of this app, that means 100 students all sharing images with each other. That’s fine for me, since I’ve never taught a class that big, but I can’t just let everyone use my copy of the app since then I’ll hit that limit pretty quick. The good news is that if you follow the steps in this section you can have your own version up and running that’ll have its own 100-user limit.

So: Step 1, sign up for a pusher account. I described how in this post.

On to step 2. Make your own copy of this google sheet. Make a google folder to hold the images and make sure its contents are viewable to the world. Then follow the instructions in this vid (:

And you’re done! Have fun!

To start the next class all you have to do is clear out the spreadsheet. You can delete the files in the image folder too if you want, but you wouldn’t have to.

I think my ideal situation is when I ask students to work on something, either individually or in groups, and then letting them all review everyone’s work. Without this app I’ve done that in the past by gathering their personal whiteboards and showing the class what I’ve found or asking them to walk around and look at other groups’ whiteboards. This way they see everyone’s and they can zoom in all they want without trying to see past people. It’s somewhat anonymous since the system doesn’t capture who uploads the image (actually it does capture the email and the date but I haven’t bothered to use that info in the UI).

Here’s an example: “Ok everyone, write down something you know about the dot product. When you’re done, submit your photo.” It seems to me you’d get some cosine (ugh) thoughts, some vectors saying “hey you, how much of you is parallel to me”, and some component-by-component scribbles. Imagine the students scrolling through all of that and then asking what patterns they’re seeing!

I also think this could be interesting in other situations. Imagine if your students were out and about doing some observations or something. They could stay on top of what everyone is seeing with this app. In other words, they don’t have to be in the same room to use the app.

What’s stopped me in the past from making this app is the annoying habit of phone manufacturers where they make their cameras have incredible pixel densities. That means that the typical file sizes are multiple megabytes. That’s not really a big deal for your phone, but it means your uploads take a while and the server just gets hammered.

This time around I googled a little and found that you can resize images before uploading them. The local machine (your student’s phone) creates a much lower density copy and uploads that instead of the original. This also makes the final product much more responsive.

Next comes the hassle of uploading and saving files in google apps script. Here’s a chapter of my GAS book explaining how I do that, though I did a little different trickery due to the copy of the image that gets created by the local machine.

Next comes making a thumbnail-based image viewer. I just used what I found here. Of course I had to make it a little more dynamic, essentially recreating the thumbnails when a new image comes in. I don’t bother to make tiny images for the thumbnails, they’re just the full image shown small. But since they’re all pretty small to begin with, that works fine.

So all together then, here’s the life cycle:

- A student loads the page
- They’re shown any images that have already been uploaded.
- If they upload a new one:
- Their machine makes the low density version
- and then sends it to google.
- Google saves the image
- retrieves the download url
- saves the url in the spreadsheet (which is why the page loaded all the old ones to begin with)
- and sends that url to all connected devices through pusher

- When a pusher message is received, the student’s machine requests the new image from the google server
- Then that image is added to the array of images at the top of the page.

Some issues:

- For some crazy reason, this only works on iPhones if they use an incognito window
- If for sure everyone loads the page at the beginning of class, you could skip the spreadsheet. But if they hit reload they’d lose all the old ones
- I gather google isn’t crazy about being an image hoster. It’s possible they’ll throttle you. In my (admittedly small scale) testing, I haven’t seen that. Famous last words, maybe
- I used a cool trick to automatically resize the image and upload it after a student selects the image. There’s no separate “submit” button. I like that
- On both iPhones and android when you select a new image you get the choice of your camera to take a picture. I love that as it reduces the steps for the students.

I’d love to hear what you think. Here are some starters for you:

- I love this. What I particularly like is . . .
- I can’t believe you reinvented the wheel, especially after such great suggestions from twitter. Loser
- I know why it has to use an incognito window on the iPhone . . .
- What’s wrong with cosine?
- Google here: cease and desist
- pusher here: are you ever going to pay for our service (note, I have paid when my school has scaled up some of my projects – they let you go month to month which is great)
- Here’s another cool way you could use it . . .
- I don’t think you need to be so worried about the original sized images and here’s why . . .

After the first day of post-it notes I volunteered to get us set up with one of the many systems that exist that do this. I went back to my hotel room and did the usual google searching, finding tools I had used (including the one that the Physics Education Research Conference organizers used earlier this month at their opening session) and others I hadn’t heard of. Unfortunately it seemed that most had a free level that only worked with 7 participants but it got expensive after that. Also most of the tools offered all kinds of extra bells and whistles, like tying in content or finding other ways to foster community, that we didn’t need.

I knew I could use the Q&A feature built into Google Slides (don’t know about that? Check it out!) but I knew there were two major issues: 1) it’s not really built to be open all day. In my experimentation in the past I noticed that the URL changes when you go back to the Q&A window a few hours later. Really that’s because it’s meant to be used in a typical 1-2 hour lecture, rather than a place to keep questions for a longer time. 2) while anyone can submit questions, you have to be logged into your Google account in order to be able to vote on questions. I’m not sure that’s a huge deal, but it always gives me pause when I’m dealing with an audience from diverse places. Certainly since my school has all our email powered by Google, it would be my go-to choice. Really it’s the first issue above that caused me to go down a different road for this workshop.

So as I was taking a shower after that disappointing google searching, it occurred to me that I might have the skills necessary to write my own solution. This post is about that.

Let’s first see it in action (there’s no sound, just trying to show all the features):

What are you seeing?

- A big box at the top to add your question.
- It has a placeholder encouraging you to type in your question. I find this much cleaner than a label above the box.
- It clears the box out after the submission has been accepted (typically ~2 seconds)

- A ranked list of questions that gets updated in real time
- If the question is from you, you can’t “upvote” it
- You can only vote once on a question
- You can’t downvote or unvote a question

- A quick description at the bottom indicating the limitations

If you care, the rest of this post explains how you could build this yourself. Why not just use mine? Because of the limitations listed at the bottom of the vid: it can only handle 100 simultaneous users. That’s totally fine for this workshop, but I can’t just provide this tool to everyone. Instead, all you need is:

- A google account (you need one, your users don’t)
- A pusher.com free-level account (this enables the webSocket technology and has the 100 simultaneous user limitation)
- A burning desire to help your fellow human beings

- Sign up for a pusher.com account, specifying the “sandbox” plan.
- Under “Channels” create an app

- Hit “Create app”
- Navigate to the new app and click on “App Keys”

- You’re done with pusher!
- Now on to setting up your google apps script

You only two tabs:

- “questions”
- id
- question
- date

- “votes”
- id
- date

Then open the App script window:

In the app script window you should have a code.gs file. Add a main.html file and a pusher.gs file:

Here’s the code for Code.gs:

```
var funcs=[];
function doGet()
{
var ss=SpreadsheetApp.getActive();
var sheet=ss.getSheetByName("questions");
var data=sheet.getDataRange().getValues();
data.shift();
var vsheet=ss.getSheetByName("votes");
var vdata=vsheet.getDataRange().getValues();
vdata.shift();
var vobj={};
data.forEach(d=>vobj[d[0]]=0);
vdata.forEach(v=>
{
// if(!vobj[v[0]]) vobj[v[0]]=0;
vobj[v[0]]++
})
var arr=data.map(m=>[m[0],m[1],vobj[m[0]]]);
var t=HtmlService.createTemplateFromFile("main");
t.funcs=funcs;
t.funcnames=t.funcs.map(f=>f.name);
t.globals={data:data, votes:vobj, arr:arr, myvotes:[]};
return t.evaluate().setTitle("Sorted Questions");
}
const init=()=>{
var cook=document.cookie;
var str=cook.match(/votes=(.*)/);
if (!str)
{
var list=[];
} else {
var list=str[1].split(",").map(m=>Number(m));
}
myvotes=[...myvotes,...list];
displayQs();
}
funcs.push(init)
const displayQs=()=>
{
arr.sort((a,b)=>b[2]-a[2]);
// html=`<h1>Sorted Questions</h1>`;
// html+=`<div><textarea rows="4" cols="50" id="newQuestion" placeholder="enter your question"></textarea></div><div><button onclick="newQlocal()">submit</button></div><hr/>`;
var html="";
html+=arr.map(a=>
{
var h=`<p>(${a[2]} votes) ${a[1]}`;
if (!myvotes.includes(a[0]))
{
h+=` <button onclick="vote(this, ${a[0]})">upvote</button>`;
} else {
h+=` You upvoted`
}
h+="</p>";
return h;
}).join(" ");
document.getElementById("main").innerHTML=html;
}
funcs.push(displayQs);
const newQlocal=()=>
{
var question=document.getElementById("newQuestion").value;
google.script.run.withSuccessHandler(newQBack).sendNewQuestion(question);
}
funcs.push(newQlocal);
const newQBack=(id)=>
{
myvotes.push(id);
var str=myvotes.join(",");
document.cookie = "votes="+str+"; SameSite=none; secure";
document.getElementById("newQuestion").value="";
displayQs();
}
funcs.push(newQBack);
const vote=(el,id)=>
{
// myvotes.push(id);
google.script.run.withSuccessHandler(newVBack).sendVote(id);
}
funcs.push(vote);
const newVBack=(id)=>
{
myvotes.push(id);
var str=myvotes.join(",");
document.cookie = "votes="+str+"; SameSite=none; secure";
displayQs();
}
funcs.push(newVBack);
function sendNewQuestion(question)
{
var sheet=SpreadsheetApp.getActive().getSheetByName("questions");
var data=sheet.getDataRange().getValues();
data.shift();
if (data.length==0)
{
var newId=1;
} else {
var newId=Math.max(...data.map(m=>m[0]))+1;
}
Logger.log(`new id is ${newId}`)
var d=new Date();
sheet.appendRow([newId,question,d]);
sendToPusher("newQ", {row: [newId,question,0]});
return newId
}
function sendVote(id)
{
var sheet=SpreadsheetApp.getActive().getSheetByName("votes");
var d=new Date();
sheet.appendRow([id,d]);
sendToPusher("newV", {id:id});
return id;
}
```

Here’s the html for main.html:

```
<!DOCTYPE html>
<html>
<head>
<base target="_top">
<link href="https://cdn.jsdelivr.net/npm/bootstrap@5.0.2/dist/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-EVSTQN3/azprG1Anm3QDgpJLIm9Nao0Yz1ztcQTwFspd3yD65VohhpuuCOmLASjC" crossorigin="anonymous">
<script src="https://js.pusher.com/5.1/pusher.min.js"></script>
</head>
<body onload="init()">
<div class="container">
<div>
<h1>Sorted Questions</h1>
<div><textarea rows="4" cols="50" id="newQuestion" placeholder="enter your question"></textarea></div><div><button onclick="newQlocal()">submit</button></div><hr/>
</div>
<div id="main">
</div>
<div>
<hr/>
<p>If your vote isn't registered after ~10 seconds, it means
the system lost your vote. Feel free to try again. The system
has a limit of 30 simultaneous votes and 100 connected users.</p>
</div>
<script>
var globals = <?!= JSON.stringify(globals) ?>;
Object.keys(globals).forEach(key=>window[key]=globals[key]);
var funcnames=<?!= JSON.stringify(funcnames) ?>;
var funcs=[<?!= funcs ?>];
funcnames.forEach((fn,i)=>window[fn]=funcs[i]);
var pusher = new Pusher(key, {
cluster: 'us3',
forceTLS: true
});
var channel = pusher.subscribe('my-channel');
channel.bind('newQ', function(data) {
arr.push(data.row)
displayQs();
});
channel.bind('newV', function(data)
{
var row=arr.find(f=>f[0]==data.id);
row[2]++;
displayQs();
})
</script>
</body>
</html>
```

And here’s the code for pusher.gs:

```
var app_id = "YOUR APP_ID HERE";
var key = "YOUR KEY HERE";
var secret = "YOUR SECRET HERE";
var cluster = "YOUR CLUSTER HERE";
function sendToPusher(event,data) {
var pvals={
appId: app_id,
key: key,
secret: secret,
cluster: cluster,
encrypted: true
};
var url = `https://api-${pvals["cluster"]}.pusher.com/apps/${pvals["appId"]}/events`;
var body = {"name":event,"channels":["my-channel"],"data":JSON.stringify(data)};
var bodystring = JSON.stringify(body);
var now=new Date();
var d = Math.round(now.getTime() / 1000);
var auth_timestamp = d;
var auth_version = '1.0';
var bodymd5 = byteToString(Utilities.computeDigest(Utilities.DigestAlgorithm.MD5, bodystring));
var wholething = `POST
/apps/${pvals["appId"]}/events
auth_key=${pvals["key"]}&auth_timestamp=${auth_timestamp}&auth_version=${auth_version}&body_md5=${bodymd5}`;
var wholethingencrypt = byteToString(Utilities.computeHmacSha256Signature(wholething,pvals["secret"]));
Logger.log(wholethingencrypt);
var options = {
'method' : 'post',
'contentType': 'application/json',
// Convert the JavaScript object to a JSON string.
'payload' : bodystring,
'muteHttpExceptions' : true
};
var urltry = UrlFetchApp.fetch(url+`?auth_key=${pvals["key"]}&auth_timestamp=${auth_timestamp}&auth_version=${auth_version}&body_md5=${bodymd5}&auth_signature=${wholethingencrypt}`, options);
}
function byteToString(byte) {
var signature = byte.reduce(function(str,chr){
chr = (chr < 0 ? chr + 256 : chr).toString(16);
return str + (chr.length==1?'0':'') + chr;
},'');
return signature;
}
```

Make sure you make the changes in the top 4 lines before moving forward.

Next you have to create a web app. Make sure to set it to execute as you but be available to everyone.

Now you can go to the url provided and it should be working!

When someone goes to the url, google sends them all the questions along with the vote tally for each. It also checks the local users cookies to see if they’ve supplied any votes. If they have, it doesn’t let them vote again. This prevents ballot box stuffing, but you should know that it’s pretty easily defeated using incognito windows.

The user can enter questions or “upvote” existing questions. When they do, an AJAX call is made to the google server (any time you see google.script.run… that’s what’s happening) where google either saves the new question (making sure to give it a unique id) or saves the vote. In either case it saves the timestamp.

After updating the spreadsheet, the “sendtopusher” function runs, sending along either the new question (along with its id and an initial vote count of zero) or the id of the new vote. That uses the webSocket that pusher has set up to send that info to all connected devices.

If a device receives a new question from pusher it adds it to the list and re-displays all the questions. If a new vote comes in from pusher it adds to the vote count for that id and re-displays all the questions (this also involves sorting based on the vote count so that the highest voted question is always at the top).

I’d love to hear your thoughts. Here are some starters for you:

- Why do you sometimes capitalize google and other times you don’t capitalize Google?
- This is cool, can it also . . .
- This is a rip-off of my cool idea. You can send checks to . . .
- This is dumb. How does this preserve the time honored tradition of the first person asking not so much a question as a 10 point rebuff of everything they heard?
- You really can’t draw very straight arrows. It’s almost as if you’re writing this post on a chromebook on your lap at LAX
- I can tell you made that vid on Loom. Why didn’t you just embed that instead of downloading it from loom, posting it on youtube, and then embedding that?
- I was at this workshop and I found this very useful.
- I was at this workshop and this was incredibly distracting
- Gross, I didn’t need to know that you had this idea in the shower.
- Here’s another way to protect against ballot box stuffing . . .

My first thoughts had to do with how you’d factor rolling into the typical analysis. Normally you determine the integral formula for the time to go from A to B on an arbitrary curve given by y(x):

where y’ is the slope of the curve, s is how far along the curve the bead has gone, v is how fast the bead is traveling, and KE is the kinetic energy which is usually a function of y (since you’re cashing in gravitational potential energy). So if it’s rolling without slipping, my first thought was that all I had to do was add in some rotational kinetic energy:

But then I realized that I had to know exactly where the center of mass was in order to figure out how much potential energy had been cashed in and I went down a rabbit hole.

You’ll see that at this point I jumped on twitter for the first time in a while (hey, my job is different now and a lot of what I do is untweetable, give me a break).

If you have a curvy road and you know the mathematical formula for one side (let’s say the road is going left to right along what we’ll call the “x-axis” and that it doesn’t turn back on itself so we can call it a function). Do you know the formula for the other side of the road? Is it just the same function with a shift? Nope. It took me a while to convince myself but this is the figure that sold me:

The blue curve is a pure sine function (why would I ever use cosine?). The orange curve is something like “sin(x)+0.32”. The green curve is what took me a while to derive but it’s really what enables a 0.32 diameter ball to fit between green and blue everywhere. Note that green and orange have the same amplitude and same frequency. Therefore, since they don’t overlap, the green curve is not a sinusoid.

So how do you derive the green curve? Well, here’s how I did it:

This represents a zoomed in version where locally the curve is flat. Also note that if you’re rolling up the other side (when the slope is positive) you need to make some adjustments to the signs in those equations. But that’s really all it takes. If you have a function for y(x) and you can calculate its slope at every location (y’), then you can figure out where the center of mass of the ball will be when you know the contact point with the curve.

Obviously I coded that in and ran it for a sine curve to get the figure above, but my same code would work with any (differentiable) function. Note that if the curve has a constant slope, the adjustments for the center of mass location are constant and then the other side of the road is truly just a shifted function. But that’s the only case that leads to that simple conclusion.

Ok, so now we know how to find the center of mass location when you know the contact point. It seems like we could figure out the potential energy drop (and hence the kinetic energy) since we know the vertical drop of the ball. Seems like we’d be in business! Alas, no, we’re not. The problem is the angular frequency, or in the equation above.

For rolling without slipping on a flat surface, you know that your linear speed and rotational speed are tied together, namely . Unfortunately, that’s not the case when rolling on a curved surface. This web page helped me understand this a little better. When you have a curved surface that has a local radius of curvature, you get this for :

where v is the speed of the contact point along the surface.

No big deal, right? it’s just some weird multiplier in front of the speed. That should make solving for the speed from the kinetic energy easy! Well, that’s what I thought, and certainly that’s what led to me erroneous twitter posts (if you scrolled through). Unfortunately, , you know that pesky local radius of curvature, is not easy to deal with. From wikipedia I learned that:

Ugh! Do you see that denominator?! Suddenly you need to know not just the slope of the function but its curvature as well. Let me tell you, that makes things gross.

Ok, gross maybe we can handle. We know how to calculate the kinetic energy and it’ll be all in terms of the (unknown) function, its slope, and its curvature. Maybe we can just close our eyes and throw it to Mathematica. Here’s where we’re at:

Fun right! Anyways, it’s technically all set to use the calculus of variations, but I’ve tried it, and wasn’t able to make any progress. I think the biggest problem is the y”s in there because they lead to a third order differential equation, which means I need to supply not only where to start the curve and what direction to head, but also the local curvature right there. Needless to say, I didn’t make much progress. If you have ideas, I’m all ears!

By the way, here’s what it looks like if you’re just doing a bead sliding down a wire:

Muuuccch easier, trust me. (Also note that if you thought I’d be using the word “cycloid” by now, you don’t get there this way. You only do if you swap x and y. You know an “obvious” thing surely your students would think to do.)

So I started googling. Here’s an awesome paper from 1946 that helps us put it all together. What they’re saying is that even when rolling on a curved surface, you can use as long as you’re using the speed of the center of mass, not the speed of the contact point. Alas, even though they’re always moving in parallel, they don’t have the same speed (think about going up and over a hump in a roller coaster, you’re moving faster than the contact point on the track). Note that they’re also saying that the center of mass follows the traditional brachistochrone! So what is this post all about!? Well, we want to know the shape of the track the ball is rolling on, and if you’ve read what I wrote above you’d know that’s different!

How did they prove it was the traditional curve? Because you get the very simple equation above instead of the incredibly ugly one if you use the coordinates of the center of mass and not the contact point. With that same simple equation, you get the same simple result (if you must: a cycloid).

But now we can put it all together. If I have a normal brachistochrone, I can find the curve for the ball to roll on by doing the coordinate shift in the figure above in reverse!

I know, I know, the blue path (the track for the ball to roll on) sure looks like a standard brachistochrone, but it’s not, because of what I was talking about above. Don’t believe me, let me hear it!

I don’t know why I didn’t do this last night, but here’s that same image with an added brachistochrone from the start to the finish of the track in green. See, I told you the blue curve wasn’t a brachistochrone:

Your thoughts? Here are some starters for you:

- What do you mean all you had to do was say “down” to imply gravity?
- Seriously, I have to read a whole other post of yours just to be able to read this one. No way! I’m unclicking. You can’t count my click.
- What do you have against rabbits? Why does going down their holes feel like an interminable complicated journey?
- What do you mean about your job being untweetable?
- What do you have against cosine?
- Duh, of course you needed to know about the curvature. What are you, an idiot?
- I know exactly where you made a mistake in that big ugly equation. For $5 I’ll tell you.
- Of course switching x and y is obvious. What’s you’re point?
- Hang on, this was solved back in 1946 and I had to read nearly your whole post to get there? Jerk.
- That blue curve is a brachistochrone and I’ve blogged about this a bunch. Try reading some time.

- Pick a target integer (I start at zero and move up by one in each iteration)
- Find a license plate (defined to be one with 3 integers on it like MN has)
- Find a way to insert mathematical operations before and between the numbers so that the result is your target

Here’s an example: Let’s say your target is 15. Here are a bunch of potential plates:

- 135 (1*3*5)
- 453 (45/3) note that just lumping 2 (or 3) numbers together is allowed
- 771 (7+7+1)
- 241 (2^4-1)

You get the gist.

I have three challenges for you:

- Find a plate that gives you the most possible targets.
- Find a plate that gives you the most
*consecutive*targets. - Find the target with the most plates that work

Here’s my quick stab at number 2: A plate with “123”:

- Target of 0: -1-2+3
- 1: -1*2+3
- 2: -(1^2)+3
- 3: (1^2)*3
- 4: 12/3
- 5: 1*(2+3)
- 6: 1+2+3
- 7: 1+2*3
- 8: 1*2^3
- 9: (1+2)*3 (or 12-3)

I got stumped trying to do 10.

Can you do better?

Your thoughts? Here are some starters for you:

- I love this. What I do is . . .
- This is dumb. The worst part is . . .
- Do you care if the sign of the answer is correct?
- Why don’t you code this up in
*Mathematica*to figure out 1, 2, and 3? - Is this what all Provosts do?
- Why don’t you watch the road when you ride?
- Next you’re going to tell me you factor mile marker signs.
- Hey, idiot, here’s how to get 10 with “123”
- Are roots allowed (like 3root8 would be the cube root of 8)?
- I think you shouldn’t be allowed to . . .
- I invented this years ago. Here’s the address to send all the cash you’re going to earn from this blog post . . .

So I thought it might be fun to see if I could make a book editor in Google Apps Script. That’s pretty meta, huh?

tl;dr? I made a ton of progress. Here’s a vid showing the features.

Here’s the things that I’m interested in having for a book editor:

- Editable anywhere (really I just mean that it should be browser based).
- Simple formatting (Markdown is my usual go-to)
- Ability to add and embed images easily
- Code highlighting (Markdown plugins to the rescue)
- Easy linking to other sections
- Easy mathematical typesetting (maybe not for this class, but still)
- Easy way to build shortcuts (like typing <<GAS>> to produce “Google Apps Script”)
- Available to everyone, but editable by only me

It turns out that a combination of bootstrap, mathjax, markdown-it, and highlighter cdns and built in GAS features enabled me to build all 8 in, so I’m pretty happy for the moment.

Not familiar with Markdown? It’s really just a way to type readable notes that can be rendered into decent html. It doesn’t have every bell and whistle, but it’s got enough for my taste. Here’s a great page describing the typical features.

For my dashboard project I figured out how to upload images to google drive and how to determine the url that can be put into an img tag. So I just reused that code, but augmenting it to let me put in a description that’s the default alt text to be used. I also made it so that you can browse the images you’ve already uploaded if you want to reuse them elsewhere in the book.

The other cool thing I learned how to do was to populate something in the browser’s clipboard on demand. Very cool.

I used to love the shortcuts I could create in documents. Things like \qm for “Quantum Mechanics” etc. I realized I could do that in this project too, but at first I just hard coded them into the rendering portion of the code. Basically I did a bunch of:

`displayText = displayText.replace(/<<qm>>/g, "Quantum Mechanics")`

However, I realized I could just put the pattern (/<<gm>>/g) and the replacement (Quantum Mechanics) into spreadsheet columns and then just run through as many as the user wants to add.

So now I just edit my spreadsheet with new shortcuts (I called them filters but the idea is the same) and the next time I load the editor those shortcuts are available.

GAS is really great for user authentication. This command gets you the email of the current user:

`var email = Session.getActiveUser().getEmail();`

and you can do whatever you want with that, including allowing editing vs limiting the user to viewing your book. When you deploy a Web App you can say that it executes as you but that it’s available to the world. When someone outside of your domain goes to your web site, the command above returns an empty string. But when someone in your domain goes to it, you get their email. So you could imagine lots of editors, for example.

GAS has lots of limitations. It’s not particularly fast, though once you pass your data to the user it’s really fast then. Sending data back and forth to the google spreadsheet usually takes a couple of seconds, which isn’t the end of the world given all the other features you get (for free!).

I pass to the user all the chapter and section names but not the detailed text of the sections, only sending that for the chosen section. So each time you go to a new section, it has to send an asynchronous request to google to get it. Again, ~2 seconds.

I’m assuming my book will get long enough that sending the whole text of the book will be problematic, so that’s why I only ever have the text from one section in memory.

Since it’s all being saved in a google spreadsheet, you have some fundamental limits to length. There are some conflicting sources out there, but there’s agreement that you can’t have more than 5,000,000 cells in the spreadsheet. That’s a lot of chapters and sections. There’s some sources that say no single cell can have more than 50,000 characters, but it seems that not everyone agrees. Assuming an average word length of, say, 8 characters that would mean that sections of the book would have to be less than 6,000 words. Since none of my blog posts have ever been that long, I don’t think I’m worried about that.

Of course I built this to write a book for my class, but since it’s all contained in a spreadsheet, it’s super easy to make copies! If you go to this spreadsheet and make your own copy, you too can write a book. All you’d have to do is:

- Clear out all the data (but not the top rows) in each tab. Note that column E in the “sections” tab is hidden, you’ll want to delete those too.
- Update your chapters and section numbers (watch the end of the vid linked above to see how that works)
- Go to tools->script editor
- In the script editor, update the top 2 lines of the globals.gs file
- Note that you’ll want to make a new folder for your images and set it to viewed by anyone

- Go to Deploy in the upper right
- Click on “New Deployment”
- Choose “Web App”
- Follow the instructions and deploy. You’ll be shown your new URL

I think I’ll probably use it for a lot of things. Even in classes where I’m not writing a book, I could still use it for organizing additional resources for my students.

I also think it might be really great for making manuals on how to do things.

Here are some starters for you:

- I love this, especially the part about . . .
- I hate this. Why don’t you just use . . .
- Wait, this is for the fall term? Don’t you have some final projects to grade?
- I think you should teach a class that teaches people how to make this tool that lets them write a book for a class on how to teach the class. That would be more meta.
- I think it would be cool if you could add . . .
- Markdown is just watered down . Why not use that instead?
- I’ve got a great idea for what I’d use this for . . .
- I’ve got a great idea for how you should never use this . . .

I’m teaching a course called “Introduction to Computational Data Science” this semester, just like I did last spring, and even with only two days under my belt I’m reminded how much I struggle with ‘input’ and ‘print’ commands. I think it has to do with the years I’ve spent programming *Mathematica* but using Jupyter and/or Colab feel quite similar.

So what am I referring to? The first assignment in this class (which has a programming class as a prerequisite) is for the students to make a video walking me through a python function they’ve made that takes an value and returns a list that’s dependent on whether the value is even or odd. It’s also supposed to throw an error if the value is not an integer. It’s really a quick test of their programming ability, and it lets me diagnose things quickly for those who might need a little help.

What’s the big deal, right? Well, nearly all the students do something like this:

```
def myFunc():
x=int(input("please give me a number, you wonderful stranger"))
if x%2:
for i in range(x//2+1):
print(i)
else:
for i in range(3*x):
print(i)
```

Why, you may ask? Because that’s how a lot of their text last semester encouraged them to do things and even the text I’m using, which has a lot of basic python chapters that I only use as reference, basically encourages that sort of function.

But I hate it! Ok, that’s, again, too strong. But I do have some problems with it, and it has to do with who the audience is.

Often in introductory programming courses you’re encouraged to think about a fictional client that you’re writing code for. Hence the “you wonderful stranger” joke above. And for clients like that, printing nice messages or values is quite a reasonable thing to do (hence the ‘print’ commands).

But for computational data work, my audience is often (always?), well, me! For me, just making a function that takes an argument and then later calling it with whatever argument I want works just fine:

```
def myFuncForMe(x):
if type(x) != int:
return []
if x%2==0:
return [x**2 for i in range(x//2+1)]
else:
return [x**3 for i in range(3*x)]
myFuncforMe(2)
```

Lots to unpack there:

- The function takes an argument instead of relying on “input”. That means that it can be used in a larger program
- The function always returns a list, though if it’s not an integer it’s an empty list. This should help it fit into larger program
- No print statements! This is a workhorse little function that can be called a bunch of times and it won’t clutter up your workspace.
- Beautiful list comprehensions instead of clunky for loops. Often if I’m looping through something I’m creating a new list and that’s what list comprehensions are built for

So if you’re slowly building a tool set that might let you gather and analyze big data, I don’t think you should be using “input” or “print” commands, at least not very much. They’re for debugging, sure, but if you’re using Jupyter or Colab, just start a new line of code to check stuff. Plus if you’re using those you can tell the story of what your code is doing so much better than if you use Spyder or some other IDE.

Ok, rant over. Your thoughts? Here’s some starters for you:

- I’m in this class right now and I need to go back and change my homework submission.
- I’m in this class right now and I need to know how I drop.
- I like this. While we’re at it, let’s try to keep students from using . . .
- I hate this. Don’t you realize how powerful “input” and “print” are?! For example . . .
- I like the two-audiences approach you’re taking. What I would add is . . .
- If you’re not writing code for someone else to use you can’t call yourself a programmer.
- If you’re writing code for someone else to use you can’t call yourself a programmer.
- I love Jupyter/Colab for these reasons and more . . .
- I turned my homework in last night and I assumed you’d be grading it instead of writing this drivel