In the early 80's there was an article about an artist who did 3D sculptures made of components. They were real circuits. One of them detected someone passing by and made strange noises that varied with their position and speed.
Congratulations for being so practical. This reminds me of the historic story when Edison was manufacturing light bulbs and for some reason he wanted to calculate the volume of the complex shape of a glass light bulb. Apparantly the mathematicians came up with some approximations but were not so sure about the accuracy of their result. Then Edison who was a very practical man came along and filled one glass bulb with water and he knew the volume of the water and also floated or immersed the glass bulb in water thus displacing the water according to Archemedes principle and again he found the outside volume of the bulb , hence he could find the volume of the glass used in the bulb. Mathematiics are very useful but being practical sometimes gets quicker results.
Edison was on a camping trip in 1887 near Rawlings Wyoming, Battle Lake when he dropped the tip of his bamboo fishing pole in the cap finite noticing the glowing car nixed fishing pole tip.... I've driven by the spit many times it is marked with a historical marker. Story is a bit iffy but makes for a good tail.. That's why I go fishing, to be inspired to invent the next big thing, my story to my wife and I'm sticking to it...! 😊
Effective resistance across the single resistor is 2R/M. M is the number of resistors (R) connecting to each junction. For your square mesh M=4. If the 2D mesh was actually a 3D cube, then M=6
+EscapeMCP This was a huge cube. 20 resistors across at least. So a facing side was a 20x20 array. The idea was to guess resistance from corner to the opposite corner across the cube. (min path=3 in your example) Some one had donated a thousands of preformed 1K resistors I kept growing it until my soldering hand got tired.
The infinite resistor grids (in 2D, 3D, 4D, …) are awesome to demonstrate how physical structures give rise to sums of infinite series. Conversely, as a rule of thumb, if you have a series and can think of a resistor network where resistance between some two nodes is a sum of the series, then the series converges. This can actually be mathematically proven, no less.
Dave I am a brazilian physics teacher. In Brazil most of people do not speak English. Would you mind if I use just a single of this video, around 3 minutes of it, translate your voice comments into portuguese to present it to my pupils ? I used my own voice in portuguese. Do you authorize me to use a peace of this video with this little changes ? In the end of the video, I gave all the credits to you, to your excelente Blog, I showed the URL of you blog here in UA-cam.
Hi Dave, This is fantastic (along with all your other videos). I like your comment about practicality and actually doing something as opposed to endless calculations. Cheers, Greg.
I'm going to build this at this year's Mini Maker Faire in Honolulu. I already bought 500 10K 1/4-watt metal film 1-percent resistors. It'll be interesting to see how many people will actually stop by and ask what the heck I'm doing. Have soldering iron, will travel...
Seeing you love measuring stuff, how about applying a voltage (say 10.000V) to the central resistor, and then measuring the voltage at various locations around the grid.
My own thought too. A geometric toroid. Intuitively i would think it would - then you could shrink that down to just a few resistors and solve it theoretically too.
I know this video is old, but I just recently found your channel. Amazing that the result is 2/pi. I like your practical approach. Would be interesting to see how the result changes if you gradually chop it down to smaller networks: 12x12, then 10x10, etc. Or if I get around to build one, maybe take measurements as it is being put together.
David Scott Any finitely sized network has resistances that are a truncated sum of an infinite series. It’s possible to prove that by induction. The convergence rates depend, among other things, on how many dimensions does the resistor grid have. Try going from a 2D grid to a 3D grid with roughly same number of resistors. Then a 4D grid. Then try going from a hyper cube (square, cube, …) to hyper sphere shape (circle, sphere, …). You’ll get quite a tour of various forms of infinite series, and can then ponder questions such as “does the circular grid converge similarly like a square grid would”. While doing so you’ll rediscover some cool theorems in calculus. All from a “simple” resistor network. I urge anyone who’s so inclined to try it out. Sometimes having a physical object to think about makes the math easier to understand. That’s certainly the case for me. When I took a mechanical vibrations course, in my mind I modeled everything with electrical circuits :)
As people have asked about three dimensional versions of this it occurs to me that if the lead lengths were constant, the resistance value between any two points might well be a constant percentage of the value of one resister. I initially expected the diagonal value to be 1/2 of the square root of 2 or .7, which it is nearly is. If lead lengths are all the same, then we can probably model this as a solid block of conductive material.
Really good work mate. nice to see some fellow aussies getting themselves out there. love hearing measurements in millimeters and grams saves having to try and guess the values the yanks are talking about all the time. What company are you working at currently?
I think the topological equivalent of the infinite grid would be a torus rather than a cube. Anyway, when you measure the resistance, that topology changes again so that there are also infinite zero-resistance "connections" between points at equal distances in the equivalent infinite grid.
+orbik Nicely said. If a grid of perfect zero tolerance resistors were overlayed on a toroid, would the measurement on any resistor or diagonal measurement (as dave did on 2D sheet he made) remain constant anywhere measured on the toroid? I expect it does. But can we apply a constant knowing how many resistors are in the grid and still get daves formula ?
unlost117 I think that depending on the size, it will show an not perfect value, but it will show the same error value no matter where you measure. :-)
It doesn't work. If you think of a torus of only a small number of resistors (say 9 or perhaps less), it quickly becomes clear you can't make resistors apply multiple times in a way that makes it look like an infinite grid.
My friend and I are going to totally make an infinite resistor grid out of 0Ω resistors for the lulz. It'll make a cool necklace and stump people :) (EEV isn't supposed to be -VEE backwards, is it? :P)
That Infinite Resistor Network you built is almost close enough to keep Rhode Island mosquitoes out. And... 420 of them to make up a problem that could keep a stoned engineer busy for hours!
Write out formulas for progressively larger networks, and you’ll see that the resistance is a sum of an infinite series. And the infinite series happens to be the one whose sum is some constant times the reciprocal of pi. Ou arises in many infinite series sums with nested fractions. Conversely, you could ask how to calculate pi using nothing but a 4-function calculator. This resistor network is a physical model for that :)
Why not connect the ends of each row and column, such that electrically there would be no center. Each resistor would be attached to the same number of resistors in every direction, kind of like a scrolling game. Would this work?
I'm really glad I saw this.Brilliant. I also agree with crapcbm, a circle would be great. I'd do it if I had a big box o" resistors laying around. Cheers
Nova Fawks PI usually a comes into play in discrete systems when some quantity is effectively a sum of an infinite series (or a nested fraction). PI appears in sums or limits of quite a few series.
That reminds me of going to school . The teacher drew up a 3 D cube of 100 ohm resistors , and said that anyone in the class who could solve it , would get an " A " grade . I didn't build the thing , I just did the math ! and got an " A " grade in the class ! anyway , Cheers , take care , and have a good day !...73 Ray aka KE8CWT & PG1920311
I wonder if this is the kind of problem/solution one would find while measuring the resistance on 2 points in a flat conductor. I've thought the closest thing to an infinite grid in real world would be a grid along the surface of a sphere. What do you think?
I'm always curious what causes people to give a thumbs down to a video like this. I suppose the end of the bell curve needs data, and perhaps it's best not looked at too closely.
Would the ACTUAL formula for calculating the opposite node of a grid this be: n / (π * R * (n - 1)) Where n = Number of dimensions, and R = resistance of resistors?
I prefer less math and more practical things too :) I was thinking if anyone build it and how large and then you showed it, pretty nice :) The measurement is close enough, you don't need like 5 digits for most things anyway
The math is very nifty though, and it’s not much more than good high school level math. It’s really cool how far simple math can go in modeling fairly complex physically realizable structures.
Wouldn't you expect the tolerance to increase over the total number of resistors? If the tolerance is 1% per resistor you are using, it would be 5% if using 5 resistors?
Not necessarily, actual tolerance calculations are more complicated, but just look it this way, if you have 1% 1ohm resistors, you expect them to be within 0.99 and 1.01ohm. If you use five of them in series, your expected result would be within 4.95 and 5.05ohm, which is still 5ohm 1% tolerance
i dont think thats the same. you would simply measure the resistance between the two probes, without taking parallel paths into account. just a guess tho
365 resistors..... I'd rather do the math By the way I am reading your "PCB Design Tutorial" which is simply brilliant put it on the tube mate it will be useful to a lot of people on the tube
Haha :D ..thank you Dave! (..just watching this video now, and just 'had' to stop at plbckPos 3m20sec and coment it. Starting to watch again the rest.. )
Actually, you can build an icosahedron or any other platonic solid as an approximation of a globe. Such networks have an interesting duality property, see ua-cam.com/video/PfIsWtXWNX4/v-deo.html. This property was found by my student Martina Furrer and was later generalized in a joint paper with Norbert Hungerbühler and Simon Jantschgi: arxiv.org/pdf/1805.01380.pdf And one can of course also verify the duality theorem experimentally, as done in hsr.othello.ch/duality.pdf (in german, but see page 7).
Sorry bout the rant. I was on Facebook typing to my friends just a bit earlier and we were dissing guitarists. (I am one) We let it get a bit out of hand and I had to leak it instead of shunting it. Peace be with you my siblings in internet fun. the rugburnz 😰
Nothing\ The analysis of any particular size of it is instructive, but it gets even more instructive when you see the patterns that are a series sum, and then you can get an analytic answer for an infinite grid without building one - and you can inductively prove that the answer you get indeed represents the result for an infinite grid.
that would only be a 40 cm by 40 cm square, neglecting connecting traces. thats a pretty big pcb, but lets do a 10x10 of those squares and get 100 million lol
i created a series of videos on that topic: ua-cam.com/play/PLoGRr8ff1uXESrWh6z0BNTYpc4Y-hlBOm.html starting with EE basics. at the end also some numeric simulation with numpy.
fun fun fun. However, approach infinity at your peril ! infinity is not a number is it aka + fun - fun + fun - fun... is it. + ( fund fun+ fun ) or -(-fun +fun -fun). ! ! ! ! IDK 1fun+2fun+3fun+4fun... = -1/12 fun = LESS THAN NO FUN @ ALL. math is easy. engineering is hard! AKA @ the end of the day something needs to get built. AKA when the arrogant lead guitarist's $13000 hand soldered by children paid $7.89 per day tag board tube amp doesn't work it is YOUR FAULT. It is NOT at all the sales rep's that told him UNDERATED output transformers and 25watt super Distortionated double doped SELLestions are conservatively rated! "Make it work nerd!" is his polite response. OH SH!T I IS ON A RANT I'll stop sorry.
![Lp. Сердце Вселенной #60 РОЖДЕНИЕ ЛОЛОЛОШКИ [Финал] • Майнкрафт](http://i.ytimg.com/vi/YoR0pAV9FVQ/mqdefault.jpg)
"I am not a math man" Says the man with a calculator fascination.
My favorite programming language ?
Solder !
This needs to be a poster.
In the early 80's there was an article about an artist who did 3D sculptures made of components. They were real circuits. One of them detected someone passing by and made strange noises that varied with their position and speed.
Congratulations for being so practical.
This reminds me of the historic story when Edison was manufacturing light bulbs and for some reason he wanted to calculate the volume of the complex shape of a glass light bulb. Apparantly the mathematicians came up with some approximations but were not so sure about the accuracy of their result. Then Edison who was a very practical man came along and filled one glass bulb with water and he knew the volume of the water and also floated or immersed the glass bulb in water thus displacing the water according to Archemedes principle and again he found the outside volume of the bulb , hence he could find the volume of the glass used in the bulb. Mathematiics are very useful but being practical sometimes gets quicker results.
practical mathematics
Edison was on a camping trip in 1887 near Rawlings Wyoming, Battle Lake when he dropped the tip of his bamboo fishing pole in the cap finite noticing the glowing car nixed fishing pole tip.... I've driven by the spit many times it is marked with a historical marker. Story is a bit iffy but makes for a good tail..
That's why I go fishing, to be inspired to invent the next big thing, my story to my wife and I'm sticking to it...!
😊
The definition of applied mathematics.
Dave could not resist to do it :)
And, we haven't the capacitance to resist the clickbait.
Effective resistance across the single resistor is 2R/M. M is the number of resistors (R) connecting to each junction. For your square mesh M=4. If the 2D mesh was actually a 3D cube, then M=6
I made a 3D version using 1K resistors. The resistance across diagonal corners is 1K.
+K6EEP Diagonal on the face (i.e. min path=2), or across the cube (min path=3)??
+EscapeMCP This was a huge cube. 20 resistors across at least. So a facing side was a 20x20 array. The idea was to guess resistance from corner to the opposite corner across the cube. (min path=3 in your example) Some one had donated a thousands of preformed 1K resistors I kept growing it until my soldering hand got tired.
It made it onto the Hack-A-Day website, so a huge influx of new viewers and subscribers!
The infinite resistor grids (in 2D, 3D, 4D, …) are awesome to demonstrate how physical structures give rise to sums of infinite series. Conversely, as a rule of thumb, if you have a series and can think of a resistor network where resistance between some two nodes is a sum of the series, then the series converges. This can actually be mathematically proven, no less.
Dave
I am a brazilian physics teacher.
In Brazil most of people do not speak English.
Would you mind if I use just a single of this video, around 3 minutes of it, translate your voice comments into portuguese to present it to my pupils ? I used my own voice in portuguese.
Do you authorize me to use a peace of this video with this little changes ?
In the end of the video, I gave all the credits to you, to your excelente Blog, I showed the URL of you blog here in UA-cam.
Beautiful, elegant, scientific. This is how an engineer solves math puzzles.
Hi Dave, This is fantastic (along with all your other videos). I like your comment about practicality and actually doing something as opposed to endless calculations.
Cheers,
Greg.
I'm going to build this at this year's Mini Maker Faire in Honolulu. I already bought 500 10K 1/4-watt metal film 1-percent resistors. It'll be interesting to see how many people will actually stop by and ask what the heck I'm doing. Have soldering iron, will travel...
Seeing you love measuring stuff, how about applying a voltage (say 10.000V) to the central resistor, and then measuring the voltage at various locations around the grid.
My own thought too. A geometric toroid. Intuitively i would think it would - then you could shrink that down to just a few resistors and solve it theoretically too.
I know this video is old, but I just recently found your channel. Amazing that the result is 2/pi. I like your practical approach. Would be interesting to see how the result changes if you gradually chop it down to smaller networks: 12x12, then 10x10, etc. Or if I get around to build one, maybe take measurements as it is being put together.
David Scott Any finitely sized network has resistances that are a truncated sum of an infinite series. It’s possible to prove that by induction. The convergence rates depend, among other things, on how many dimensions does the resistor grid have. Try going from a 2D grid to a 3D grid with roughly same number of resistors. Then a 4D grid. Then try going from a hyper cube (square, cube, …) to hyper sphere shape (circle, sphere, …). You’ll get quite a tour of various forms of infinite series, and can then ponder questions such as “does the circular grid converge similarly like a square grid would”. While doing so you’ll rediscover some cool theorems in calculus. All from a “simple” resistor network. I urge anyone who’s so inclined to try it out. Sometimes having a physical object to think about makes the math easier to understand. That’s certainly the case for me. When I took a mechanical vibrations course, in my mind I modeled everything with electrical circuits :)
It would be really interesting to see reading as the grib was being built to show the convergence of the solution as every new row/column is added
The pubs weren't open at the time right?
As people have asked about three dimensional versions of this it occurs to me that if the lead lengths were constant, the resistance value between any two points might well be a constant percentage of the value of one resister. I initially expected the diagonal value to be 1/2 of the square root of 2 or .7, which it is nearly is. If lead lengths are all the same, then we can probably model this as a solid block of conductive material.
This is definitely taking practicality to its limits.
Could you please describe how to derive equivalent resistance for diagonal case?
How sturdy is that grid? If you want to keep it for decoration, maybe you can also use it to hang other components onto to add a bit of variation.
Somehow I want to make a giant version of this with 0201 resistors and then put it under a glass table.
420 resistors? now we know how you came up with the idea
Yea, I was scratching my head about that one too. lol
Really good work mate.
nice to see some fellow aussies getting themselves out there. love hearing measurements in millimeters and grams saves having to try and guess the values the yanks are talking about all the time.
What company are you working at currently?
Good video. What about making it in to a cube?
I call this, "Dave's Tholian Web Episode."
Behold: the Jones Resistive Gridiron!
Nice work,
plus more than 50 "you know"s in this video
:)
What would it be in 3 dimensions? 4?
@@pahom2 if "R" is the resistance on each edge, then the resistance between nearest neighbor vertices in a "D"-dimensional hypercube lattice is $R/D$.
I want to see this built with a million 0402 SMT packages :P
Pick and Place workout much? :D
could you build a cube out of six of those and get the infinite properties?
I think the topological equivalent of the infinite grid would be a torus rather than a cube. Anyway, when you measure the resistance, that topology changes again so that there are also infinite zero-resistance "connections" between points at equal distances in the equivalent infinite grid.
An ifinite resistor torus with Rodin coil alignment would surely wet daves practical appetite and would look stunning
+orbik Nicely said. If a grid of perfect zero tolerance resistors were overlayed on a toroid, would the measurement on any resistor or diagonal measurement (as dave did on 2D sheet he made) remain constant anywhere measured on the toroid? I expect it does. But can we apply a constant knowing how many resistors are in the grid and still get daves formula ?
unlost117 I think that depending on the size, it will show an not perfect value, but it will show the same error value no matter where you measure. :-)
It doesn't work. If you think of a torus of only a small number of resistors (say 9 or perhaps less), it quickly becomes clear you can't make resistors apply multiple times in a way that makes it look like an infinite grid.
My friend and I are going to totally make an infinite resistor grid out of 0Ω resistors for the lulz. It'll make a cool necklace and stump people :)
(EEV isn't supposed to be -VEE backwards, is it? :P)
This was a great demo.
Wouldn't it be easier to use an electrolyte solution for testing? Maybe salt water.
That Infinite Resistor Network you built is almost close enough to keep Rhode Island mosquitoes out. And... 420 of them to make up a problem that could keep a stoned engineer busy for hours!
how did a pi come into this?
i just cant picture where the number came from
Write out formulas for progressively larger networks, and you’ll see that the resistance is a sum of an infinite series. And the infinite series happens to be the one whose sum is some constant times the reciprocal of pi. Ou arises in many infinite series sums with nested fractions. Conversely, you could ask how to calculate pi using nothing but a 4-function calculator. This resistor network is a physical model for that :)
Why not connect the ends of each row and column, such that electrically there would be no center. Each resistor would be attached to the same number of resistors in every direction, kind of like a scrolling game. Would this work?
I'm really glad I saw this.Brilliant. I also agree with crapcbm, a circle would be great. I'd do it if I had a big box o" resistors laying around. Cheers
I'm new to EE so I'm confused. How does pi come in to play with 2 of them?
Nova Fawks PI usually a comes into play in discrete systems when some quantity is effectively a sum of an infinite series (or a nested fraction). PI appears in sums or limits of quite a few series.
That reminds me of going to school . The teacher drew up a 3 D cube of 100 ohm resistors , and said that anyone in the class who could solve it , would get an " A " grade . I didn't build the thing , I just did the math ! and got an " A " grade in the class ! anyway ,
Cheers , take care , and have a good day !...73
Ray aka KE8CWT & PG1920311
Array For Dave!!
Haha, nice work!
I wonder if this is the kind of problem/solution one would find while measuring the resistance on 2 points in a flat conductor.
I've thought the closest thing to an infinite grid in real world would be a grid along the surface of a sphere. What do you think?
What would happen if you made a 3d shape like a cube?
What did he exactly say after .5 an ohm? @1:15
did he say "not .5 *R "
"naught" is another work for zero. wikipedia.org/wiki/0
I'm always curious what causes people to give a thumbs down to a video like this. I suppose the end of the bell curve needs data, and perhaps it's best not looked at too closely.
Use LTspice to get answer quickly.
Would the ACTUAL formula for calculating the opposite node of a grid this be:
n / (π * R * (n - 1))
Where n = Number of dimensions, and R = resistance of resistors?
what about a spherical infinite resistor dave?
It is irresistible :-)
I prefer less math and more practical things too :) I was thinking if anyone build it and how large and then you showed it, pretty nice :)
The measurement is close enough, you don't need like 5 digits for most things anyway
The math is very nifty though, and it’s not much more than good high school level math. It’s really cool how far simple math can go in modeling fairly complex physically realizable structures.
Wouldn't you expect the tolerance to increase over the total number of resistors? If the tolerance is 1% per resistor you are using, it would be 5% if using 5 resistors?
Not if the tolerance is normally distributed. If its skewed or rounded, you would be correct.
Not necessarily, actual tolerance calculations are more complicated, but just look it this way, if you have 1% 1ohm resistors, you expect them to be within 0.99 and 1.01ohm. If you use five of them in series, your expected result would be within 4.95 and 5.05ohm, which is still 5ohm 1% tolerance
+Daniel T sadly most companies have a tendency to lean to being high or low still writing spec
Great Video!!
I wondered anyone done math, can they post solution and derivation?
ok dave, 420 resistors... we get it you vape bro :P
the mix of references in this burns...
How about if the outer most resistors on the perimeter were 5K to 'terminate' the grid since that's what the expected resistance would be.
i dont think thats the same. you would simply measure the resistance between the two probes, without taking parallel paths into account. just a guess tho
Hello from the future xD
What about a cap's grid measures?? in a cubic form, I mean 6 sides
AHahahha!
What about a cube? :)
Hehehehe, I like this one, nice puzzle and excellent answer to it :)
Now send it to Photonicinduction and tell him to make some HV barbecue cooking with it! Just imagine cooking a steak on this! :D
@chandin69 pi crops up in all sorts of apparently unlikely situations. It's really quite remarkable. Read "The Life Of Pi", it's a fascinating read.
Question is asked, the person opens up a briefcase and pulls out the resistor network and a multimeter...
365 resistors.....
I'd rather do the math
By the way I am reading your "PCB Design Tutorial"
which is simply brilliant
put it on the tube mate
it will be useful to a lot of people on the tube
please revisit the topic after so many years of youtube!
Haha :D ..thank you Dave!
(..just watching this video now, and just 'had' to stop at plbckPos 3m20sec and coment it. Starting to watch again the rest.. )
+RolfT S --> And..Yes, frame it :) ..its electronics-art
Infinite capacitor network next please!
because shorted out caps will be so much fun? ;)
I watched the #1000th EEVBlog Video, then the UA-cam suggest me this video...coincidence?
great vid, so funny xD
Would be interesting to make cylinder or globe :)
Actually, you can build an icosahedron or any other platonic solid as an approximation of a globe. Such networks have an interesting duality property, see ua-cam.com/video/PfIsWtXWNX4/v-deo.html. This property was found by my student Martina Furrer and was later generalized in a joint paper with Norbert Hungerbühler and Simon Jantschgi: arxiv.org/pdf/1805.01380.pdf
And one can of course also verify the duality theorem experimentally, as done in hsr.othello.ch/duality.pdf (in german, but see page 7).
@GRAHAMAUS i remember reading "The Life Of Pi" about 5 years ago, and it was about some dude lost at sea .. nothing to do with math
Cool! 420 Resistors :-)
For my next trick I present - Schrodinger's cat! Urgh!
Sorry bout the rant. I was on Facebook typing to my friends just a bit earlier and we were dissing guitarists. (I am one) We let it get a bit out of hand and I had to leak it instead of shunting it. Peace be with you my siblings in internet fun. the rugburnz 😰
He said "naught", as in zero.
Thanks much :)
Good. Now let's analyze the whole circuit if you put 5v on one corner and ground the other. Every single junction for current and voltage drop.
Nothing\ The analysis of any particular size of it is instructive, but it gets even more instructive when you see the patterns that are a series sum, and then you can get an analytic answer for an infinite grid without building one - and you can inductively prove that the answer you get indeed represents the result for an infinite grid.
now i want the maths!
Classic!
Never imagined he was a pothead! lol
Forget math!
Build bigger resistor array!
you know, I counted more than 30 "you know"s half way to the video :)
that would only be a 40 cm by 40 cm square, neglecting connecting traces. thats a pretty big pcb, but lets do a 10x10 of those squares and get 100 million lol
Interesting... I wonder what would happen if instead of building it as a 2 dimensional grid if you made it a 3 dimensional cube. Could be interesting.
You'd want to get all Buckminster Fuller and build a geodesic form.
I'm your host, Dave fucking Jones.
You can tell you're not s mathematician. If you were you'd have made a 3D representation of a 4 dimensional resistor network!
Interesting.
Also 51,000 view, noice.
But your expeimental method soes not give you the theory behind the mechanics of the problem... Though I like your arts skills!
i created a series of videos on that topic: ua-cam.com/play/PLoGRr8ff1uXESrWh6z0BNTYpc4Y-hlBOm.html starting with EE basics. at the end also some numeric simulation with numpy.
Do the math please :D
hah, beautiful
fun fun fun. However, approach infinity at your peril ! infinity is not a number is it aka + fun - fun + fun - fun... is it. + ( fund fun+ fun ) or -(-fun +fun -fun). ! ! ! ! IDK 1fun+2fun+3fun+4fun... = -1/12 fun = LESS THAN NO FUN @ ALL. math is easy. engineering is hard! AKA @ the end of the day something needs to get built. AKA when the arrogant lead guitarist's $13000 hand soldered by children paid $7.89 per day tag board tube amp doesn't work it is YOUR FAULT. It is NOT at all the sales rep's that told him UNDERATED output transformers and 25watt super Distortionated double doped SELLestions are conservatively rated! "Make it work nerd!" is his polite response. OH SH!T I IS ON A RANT I'll stop sorry.
#420
87th!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!