Oh shit Staecker has a new video on something, and it's a grid of dots, and I am HERE for it. Analog calculation devices are so weirdly fascinating. I'm a board game enthusiast and there's a small part of me that thinks there's a bit of overlap there, board games are also a kind of supremely weird analog calculator sometimes. Like this dot planimeter, so simple you could use it as a component for some kind of game where you create irregular shapes.
I can only wonder why we got to this idea so late, the only thing I can think of is that the ability to produce high precision grids on transparent material is a relatively late invention. I also suspect that demand for measurements of the area of small objects is also relatively recent. About the only thing I can think of an ancient or medieval person wanting to know the area of is land, and while you could use a dot planimeter to measure the area of a drawing of land that would rely on the drawing being accurate in the first place, and if you've made an accurate drawing of a piece of land you've already measured it. For small things I'm sure it's either the count or the weight that mattered in day-to-day life. I also noticed that this is an extrapolation of the ruler into two dimensions, and I was wondering if you could further extrapolate to three. But I suppose that's not really necessary because it's relatively easy to measure the volume of objects so long as they're waterproof.
There is another simple way to do this that was actually used in biology for cell research and such, were you simply trace it on thick card stock, cut your desired object out and simply weigh it! :)
Can confirm, it doesn't work very well for small things (too much variables in weighting and paper itself) but for something like hand-sized leaf it's pretty great!
I was going to mention something similar... one of my engineering professors described how they used this method for integrating functions before computers were available. He said he would plot it on large butcher paper for higher accuracy, and for critical applications companies used aluminum foil because of the higher density and uniformity.
@@lokalnyork hmm. maybe if you have a photo of it you can enlarge that through the magic of lenses and stuff? obviously this is all obsolete now but i'm just thinking about a biologist in a photo lab with red light making a 1000:1 print of a cell to cut out and weigh.
In my biotechnology studies we used 1x1mm graph paper (with 1x1 cm extra thick grid to save time) for area of leaves. For digital version one could use black and white picture/scan of known area and "count" black pixels with histogram in Photoshop or other equivalent.
Perhaps part of the reason the idea seems so obvious to us is that we're so used to dealing with images made of pixels, so counting squares to find areas feels natural?
I remember in grad school classes being introduced to the idea of "elegance." In linguistics, for crying out loud, there's elegance and then there's my kinda stuff: true dreck. But at least now I know that my stuff is elegant dreck. Thanks again for another beautiful video, Chris. Stay safe & stay well.
I love the simplicity of this thing! Just wanted to mention that ImageJ/Fiji is a great software if you want to do these things digitally. It's at least used a lot for quantitation of microscopy images/videos, and it can do a lot of cool stuff. No idea if there is any way of correcting for lens distortion and stuff tho.
I faintly remember seeing this in grade school when they were teaching area. I always thought it was just a teaching device and didn't expect it to be an actual tool that people used professionally.
Hey, i just discovered your channel and i really enjoy this series, thanks! From Pick's theorem, i came to a personnal reflexion: when you want to estimate an area by the dot counting method, you get a better approximation by using the formula: A = d + h/2 - 1 Where d is the number of dots in the area, and h the number of dot exactly on the line. In your example, you would have come to an approximated area of 99 the first time, 99 the 2nd time, 99 the 3rd time and 100.5 the 4th time, that to say an area of 99.375 in average.
This is a good idea- obviously Pick's theorem is relevant here. The dot planimeter with the empty/filled circles would tend to give the area as d + h/2. I'm not sure exactly how to correctly account for the extra -1 from Pick's theorem. The situation in Pick's theorem may make the dot count artificially high, since Pick's theorem assumes that the vertices are on the dot points. Positioning the dots at random on a polygon will generally miss the vertex points. So maybe the -1 works as a compensation for over-weighting the vertices. Thanks for the idea!
@@ChrisStaecker You gave me some homework here :D The extra -1 is usually related to a think called the Euler characteristic, but it doesn't look to be the case here. Assuming that the edge has a thickness t < 1, and with a probabilist approach, i come to the conclusion that it should be a -t² in our situation, so 0 is a good approximation. Considering two adjacent shapes, t² is the expectation for the two intersections of their edges to each contain a point with integer coordinates. It's obviouly 1 with Pick's hypothesis.
My math teacher told me I needed to know how to calculate the areas of different shapes for when I paint a room or buy carpet. Turns out I could have just counted the dots...
On the school, a techer show me how to measure and area of a strange sbape. Firs we cut a 1 cm2 squere of paper and mesure their weight white a scale and thwen we cut the strange shape ans mesure their mas and made a relation a d found the area.
When we were trying to measure the area of an irregular shape about 50 years ago we found an Axe Planemeter. Check it out, just as frustrating as counting dots.
So cool. Have you see the docking scope used in the lunar module? TheEngineeringGuy did a great vid on it, if you haven't seen it, I'm sure you'll admire it for similar reasons.
Just watched it- very nice! I'm sure the Apollo program and other similar megaprojects are full of custom-built one-off tools that solve problems in simple & novel ways. Mostly probably unknown to the public too! Thanks-
When did a reasonably transparent material become widely available? That seems to be the point at which this could have been invented. Did they use glassine in the old days?
@@ChrisStaecker hmmm. In that case I’m surprised someone didn’t come up with it earlier. Darwin and those other exploring naturalists carried bulkier stuff with them than a pane of dotted glass. Your videos are aces, btw. I keep Forwarding them to my mathy friends.
@@johnsrabe It would be easy to do using a quadrat (using intersection points as the dots), which is more portable and probably easier to make. Seems like those also appeared around 1900.
"You may think you could've come up with that, But you didn't" ...I... I basically did. Every time I was given a grid with a shape on it, I found it easier to count the number of squares in it for the area. If I kept doing that later in life with no better way like many people did all those years ago, I'm sure those two bits would come together in my mind. I suppose the issue is I never had plastic to lay over the shapes?
I don't know how I got here, but I'm glad I did.
Oh shit Staecker has a new video on something, and it's a grid of dots, and I am HERE for it. Analog calculation devices are so weirdly fascinating. I'm a board game enthusiast and there's a small part of me that thinks there's a bit of overlap there, board games are also a kind of supremely weird analog calculator sometimes. Like this dot planimeter, so simple you could use it as a component for some kind of game where you create irregular shapes.
Staecker new vid alert
A black paper sheet is the most accurate dot planimeter
I can only wonder why we got to this idea so late, the only thing I can think of is that the ability to produce high precision grids on transparent material is a relatively late invention. I also suspect that demand for measurements of the area of small objects is also relatively recent. About the only thing I can think of an ancient or medieval person wanting to know the area of is land, and while you could use a dot planimeter to measure the area of a drawing of land that would rely on the drawing being accurate in the first place, and if you've made an accurate drawing of a piece of land you've already measured it. For small things I'm sure it's either the count or the weight that mattered in day-to-day life.
I also noticed that this is an extrapolation of the ruler into two dimensions, and I was wondering if you could further extrapolate to three. But I suppose that's not really necessary because it's relatively easy to measure the volume of objects so long as they're waterproof.
There is another simple way to do this that was actually used in biology for cell research and such, were you simply trace it on thick card stock, cut your desired object out and simply weigh it! :)
Can confirm, it doesn't work very well for small things (too much variables in weighting and paper itself) but for something like hand-sized leaf it's pretty great!
I was going to mention something similar... one of my engineering professors described how they used this method for integrating functions before computers were available. He said he would plot it on large butcher paper for higher accuracy, and for critical applications companies used aluminum foil because of the higher density and uniformity.
@@lokalnyork hmm. maybe if you have a photo of it you can enlarge that through the magic of lenses and stuff? obviously this is all obsolete now but i'm just thinking about a biologist in a photo lab with red light making a 1000:1 print of a cell to cut out and weigh.
In school for map-work in Geography, we used to make our own dot planimeters to calculated the area of farmland. : ) Awesome videos!
By the way: there were large polar planimeters that were used to measure the size of bear skins in Russia.
I'm here for the Pupcake content
> shows Hugo Steinhaus as a German Oktoberfest Waitress
> next up is Hugo's paper in original Polish
In my biotechnology studies we used 1x1mm graph paper (with 1x1 cm extra thick grid to save time) for area of leaves. For digital version one could use black and white picture/scan of known area and "count" black pixels with histogram in Photoshop or other equivalent.
Perhaps part of the reason the idea seems so obvious to us is that we're so used to dealing with images made of pixels, so counting squares to find areas feels natural?
I remember in grad school classes being introduced to the idea of "elegance." In linguistics, for crying out loud, there's elegance and then there's my kinda stuff: true dreck. But at least now I know that my stuff is elegant dreck.
Thanks again for another beautiful video, Chris. Stay safe & stay well.
I love the simplicity of this thing! Just wanted to mention that ImageJ/Fiji is a great software if you want to do these things digitally. It's at least used a lot for quantitation of microscopy images/videos, and it can do a lot of cool stuff. No idea if there is any way of correcting for lens distortion and stuff tho.
I faintly remember seeing this in grade school when they were teaching area. I always thought it was just a teaching device and didn't expect it to be an actual tool that people used professionally.
Hey, i just discovered your channel and i really enjoy this series, thanks!
From Pick's theorem, i came to a personnal reflexion: when you want to estimate an area by the dot counting method, you get a better approximation by using the formula:
A = d + h/2 - 1
Where d is the number of dots in the area, and h the number of dot exactly on the line. In your example, you would have come to an approximated area of 99 the first time, 99 the 2nd time, 99 the 3rd time and 100.5 the 4th time, that to say an area of 99.375 in average.
This is a good idea- obviously Pick's theorem is relevant here. The dot planimeter with the empty/filled circles would tend to give the area as d + h/2. I'm not sure exactly how to correctly account for the extra -1 from Pick's theorem.
The situation in Pick's theorem may make the dot count artificially high, since Pick's theorem assumes that the vertices are on the dot points. Positioning the dots at random on a polygon will generally miss the vertex points. So maybe the -1 works as a compensation for over-weighting the vertices. Thanks for the idea!
@@ChrisStaecker You gave me some homework here :D
The extra -1 is usually related to a think called the Euler characteristic, but it doesn't look to be the case here. Assuming that the edge has a thickness t < 1, and with a probabilist approach, i come to the conclusion that it should be a -t² in our situation, so 0 is a good approximation.
Considering two adjacent shapes, t² is the expectation for the two intersections of their edges to each contain a point with integer coordinates. It's obviouly 1 with Pick's hypothesis.
A flatbed scanner and an edge detection algorithm could be an overcomplicated electronic dot planimeter
I just draw these odd shapes in a cad program then let the computer tell me the area.
OR we can make like Archimedes and just sum with a Method of Exhaustion...
This would be very useful with a tighter grid and a dot counter app that you can get on your phone (I could only find one app and it's $1.99).
using a cell phone app to count the dots on an old school Dot Planimeter is a very satisfying idea
If you scan it and count the pixels, You're using a dot planimeter with extra steps.
We also use these in geology for calculating the area percentage of minerals in rock thin sections.
I vaguely remember using these at some point in my early school career. Pretty sure it was elementary or middle school.
My math teacher told me I needed to know how to calculate the areas of different shapes for when I paint a room or buy carpet. Turns out I could have just counted the dots...
I dont know why but those pages in polish scared the hell out of me.
the guy who invented the planimeter must have been so mad when his expensive ass tool is made irrelevant by a piece of paper w dots on it
this channel is so criminally underrated.
On the school, a techer show me how to measure and area of a strange sbape. Firs we cut a 1 cm2 squere of paper and mesure their weight white a scale and thwen we cut the strange shape ans mesure their mas and made a relation a d found the area.
That squares with a dot in the center is how I measure distances in a grid.
what the heck why is this video so good I have negative interest in math stuff
analog computers
When we were trying to measure the area of an irregular shape about 50 years ago we found an Axe Planemeter. Check it out, just as frustrating as counting dots.
Yes- future video!
So cool. Have you see the docking scope used in the lunar module? TheEngineeringGuy did a great vid on it, if you haven't seen it, I'm sure you'll admire it for similar reasons.
Just watched it- very nice! I'm sure the Apollo program and other similar megaprojects are full of custom-built one-off tools that solve problems in simple & novel ways. Mostly probably unknown to the public too! Thanks-
grids of dots are boring, I’m only going to watch a minute or two and then move on to something else more interesting…
That's what I said when I first heard about this thing.
inside one dot
love it
When did a reasonably transparent material become widely available? That seems to be the point at which this could have been invented. Did they use glassine in the old days?
Could use real glass I suppose.
@@ChrisStaecker hmmm. In that case I’m surprised someone didn’t come up with it earlier. Darwin and those other exploring naturalists carried bulkier stuff with them than a pane of dotted glass. Your videos are aces, btw. I keep
Forwarding them to my mathy friends.
@@johnsrabe It would be easy to do using a quadrat (using intersection points as the dots), which is more portable and probably easier to make. Seems like those also appeared around 1900.
AAAA+++. Will watch again!
"You may think you could've come up with that, But you didn't"
...I... I basically did. Every time I was given a grid with a shape on it, I found it easier to count the number of squares in it for the area. If I kept doing that later in life with no better way like many people did all those years ago, I'm sure those two bits would come together in my mind. I suppose the issue is I never had plastic to lay over the shapes?
Sounds like you did! I didn't... but I didn't really have any reason to invent an area-measuring tool either.