I am actually busy working on the 24-Cell:
https://en.wikipedia.org/wiki/24-cell
A most interesting problem. Has no analog in 3D!
https://en.wikipedia.org/wiki/24-cell
A most interesting problem. Has no analog in 3D!
Four-dimensional geometry is Euclidean geometry extended into one additional dimension.
Illustration:
2D - square
3D - cube
4D - tesseract
Just as the dimension of depth can be added to a square to create a cube, a fourth dimension can be added to a cube to create a tesseract.
Indeed, we are in a 4D universe.
This is why your GPS needs to receive FOUR satellites to figure where you are.
A byproduct of the calculation giving your coordinates ( latitude, longitude and altitude ) is your time with a tremendous accuracy.
The accuracy of the atomic clocks onboard the satellites that get regular adjustments from ground bases holding atomic clocks of the very best accuracy.
These adjustments because of clock drifts from gravity and speed. Without these two Einsteinien phenomenons, GPS would simply not work at all, because the clocks drifts would quickly induce way too wrong results.
GPS is wonderful.
For those who can't get their heads round Einsteinian four-dimensional spacetime, here's a simplification.
You need four intersecting spheres to exactly pinpoint your location, and hence you need FOUR satellites (see attachment).
I remember you, mchambin, saying that this was the basis of early GPS algorithms.
You need four intersecting spheres to exactly pinpoint your location, and hence you need FOUR satellites (see attachment).
I remember you, mchambin, saying that this was the basis of early GPS algorithms.
Attachments
T
It is not needed to determine which of the two points is the right one.
The true reason is. You do not have the time of the receiver accurate enough. You cannot afford the cost and the weight and size of an atomic clock, at the receiver even an expensive crystal clock would not be good enough.
So, the receiver's time is unknown.
Thanks to a 4th satellite you can workaround this issue, calculating all 4 coordinates x,y,z,t ( receiver location: x,y,z, receiver time: t ).
A neat algorithm to do this is Bancroft method. It gives the answer with 4 satellites and with more than 4 which is usually the case.
https://www.google.com/url?sa=t&sou...EQFnoECAwQAQ&usg=AOvVaw24LtcQj3F-rX8WIQ2juxMw.
This is THE wrong reason for a 4th satellite.
It is not needed to determine which of the two points is the right one.
The true reason is. You do not have the time of the receiver accurate enough. You cannot afford the cost and the weight and size of an atomic clock, at the receiver even an expensive crystal clock would not be good enough.
So, the receiver's time is unknown.
Thanks to a 4th satellite you can workaround this issue, calculating all 4 coordinates x,y,z,t ( receiver location: x,y,z, receiver time: t ).
A neat algorithm to do this is Bancroft method. It gives the answer with 4 satellites and with more than 4 which is usually the case.
https://www.google.com/url?sa=t&sou...EQFnoECAwQAQ&usg=AOvVaw24LtcQj3F-rX8WIQ2juxMw.
I wrote how GPS works over in another thread:
"To calculate the time the GPS signals take to arrive, the GPS receiver needs to know the time very accurately. Rather than have to install an expensive atomic clock within the receiver itself, a signal from a fourth satellite is used by the receiver to solve an equation that lets it determine the exact time.
If the GPS receiver is only able to get signals from 3 satellites, you can still get your position, but it will be less accurate. The GPS receiver needs 4 satellites to work out your position in 3 dimensions, including altitude. If only 3 satellites are available, the GPS receiver can get an approximate position by making the assumption that you are at mean sea level."
"To calculate the time the GPS signals take to arrive, the GPS receiver needs to know the time very accurately. Rather than have to install an expensive atomic clock within the receiver itself, a signal from a fourth satellite is used by the receiver to solve an equation that lets it determine the exact time.
If the GPS receiver is only able to get signals from 3 satellites, you can still get your position, but it will be less accurate. The GPS receiver needs 4 satellites to work out your position in 3 dimensions, including altitude. If only 3 satellites are available, the GPS receiver can get an approximate position by making the assumption that you are at mean sea level."
It was you, mchambin, who introduced me to the Bancroft Method.
Get your teeth into this, you mathematicians!
https://gssc.esa.int/navipedia/index.php/Bancroft_Method
Get your teeth into this, you mathematicians!
https://gssc.esa.int/navipedia/index.php/Bancroft_Method
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