Okay, I spent too much time with fields and rings and need to spend
more time on geometry. What I hope to accomplish is (1) show the lines
of Riemann geometry are formed by the Adics (2) show the poles of the
Adics (3) show the equator line of Adics.

If we examine the first given line which is a great circle is divided
into two parts. The part of ending zeroes and the part of ending 9s in
10-adics. So the 10-adics starts with ...0000 then ....0001 and ends
with ....9999 which is -1. So we can view this as a great circle.

Now what are the other great circles in 10-adics? From what I
understand from Dik, the 10-adics integers is a ring and so would have
additive inverses. So what is the number 1/2 in 10-adics and in
11-adics? What is the number -1/2 in both 10-adics and 11-adics? Then
what is the number 1.5 in both 10-adics and 11-adics? And what is -1.5
in both 10-adics and 11-adics? I am going to use 10-adics and 11-adics
because one is a ring and the other a field and use both adic integers
and adic rationals.

So what are the great circles in adics? Well we have the first one of
ending in zeroes connected to ending in 9s which is one great circle.
Now what is a second great circle in 10-adics? It would be the
repetition of what 1/2 and -1/2 in 10-adics. Then what would be a third
great circle in 10-adics? It would be the combining of 1.5 repetition
with -1.5 repetition in 10-adics where the starting numbers have the
1.5 repetition and the ending numbers have the -1.5 repetition just as
the starting numbers have 0000s as repetition and the ending numbers
have 999s as repetition. In this manner we have an infinite number of
great circles, but we lack the equator great circle.

As for the poles can we find two unique points of intersection? I used
to think the idempotents would be such poles but the p-adics lack
idempotents. So I am baffled as to the poles would be.

However, I think I have made progress on the equator line composed of
roots of negative numbers in adics. We know that sqrt -1 for Reals acts
as a orthogonality or 90 degree angle. So the adics abound in roots of
negative numbers. And I speculate that the union of all the roots of
negative numbers in 10-adics or 11-adics is the equator line for those
adics.

So I lack the poles and need to think about it some more.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

Now I used to write it this way:

Eucl geom = Riem geom + Loba geom

Or I could write it this way

Eucl geom = Riem geom appended with Loba geom

If the native numbers of Eucl geom are the infinite strings rightward,
finite leftwards

and if the native numbers of Riem geom are infinite strings leftward,
(perhaps finite rightwards)

Then what are the native points of Lobachevskian geometry? My thinking
lately is that the Adics serve as both the native points of Riem as
well as the native points of Loba geometry. Perhaps the native points
of Riem are the Adic Integers and the native points of Loba geometry
are the Adic Rationals (those finite strings rightward)

But much more thought and reflection has to go into this before I am
certain.

I am certain that the disc in 2-D is a model of Loba geometry and that
the circle in 2-D is a model of Riem geometry. Now it does not take a
rocket scientist to understand that the circle placed inside the hole
of the disc then covers Eucl geometry so that the equation Euclid geom
= Riem geom + Loba geom holds true.

In 3-D the same happens in that a saddle shape model and a sphere
placed into the saddle have points of intersection (a snug fit so to
speak) and thus a manifold of that region of snug fit would be a
Euclidean 3-D geometry. And a trumpet shape (the musical instrument) is
a Loba model and so again we take a large enough sphere placed against
the side of a trumpet is a Eucl 3-D manifold. Another Loba model is a
Well (like a water well hole) and we can plug up the bottom of the well
by a sphere and so again a Eucl manifold is produced.

These models show us the correctness of the equation:

Eucl geom = Riem geom + Loba geom

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

wrote:
[...]
Now what are the other great circles in 10-adics? From what I
understand from Dik, the 10-adics integers is a ring and so would have
additive inverses. So what is the number 1/2 in 10-adics

1/2 is the MULTIPLICATIVE inverse of 2, not the ADDITIVE inverse of 2.

and in
11-adics? What is the number -1/2 in both 10-adics and 11-adics? [...]

Asking what the additive inverse of -1/2 is in the 10-adics is like
asking what its additive inverse is in the integers.

So I lack the poles and need to think about it some more.

Finally, something I accept as true. But if you needed to think about
it some more, why did you post this?

--- Christopher Heckman

In article .com writes:
Now what are the other great circles in 10-adics? From what I
understand from Dik, the 10-adics integers is a ring and so would have
additive inverses. So what is the number 1/2 in 10-adics and in
11-adics? What is the number -1/2 in both 10-adics and 11-adics? Then
what is the number 1.5 in both 10-adics and 11-adics? And what is -1.5
in both 10-adics and 11-adics? I am going to use 10-adics and 11-adics
because one is a ring and the other a field and use both adic integers
and adic rationals.

As you do not understand the things you are working with at all, and are
not able to do even simple calculations, I will help you. But with what
has been written before you should have been able to do the calculations
yourself ('a' is the digit '10' in the 11-adics):
10-adic 11-adic
1/2 0.5 ...55556
-1/2 ...99999.5 ...55555
3/2 1.5 ...55557
-3/2 ...99998.5 ...55554
and just for fun:
1/11 ...09091 0.1
-1/11 ...90909 ...aaaaa.a

See a pattern?

However, I think I have made progress on the equator line composed of
roots of negative numbers in adics. We know that sqrt -1 for Reals acts
as a orthogonality or 90 degree angle.

That is only a visualisation of the complex numbers, nothing more.

So I lack the poles and need to think about it some more.

The thinking part is right on the dot. But before thinking comes
understanding what you are playing with, and you are still lacking that.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Dik T. Winter wrote:
In article .com writes:
Now what are the other great circles in 10-adics? From what I
understand from Dik, the 10-adics integers is a ring and so would have
additive inverses. So what is the number 1/2 in 10-adics and in
11-adics? What is the number -1/2 in both 10-adics and 11-adics? Then
what is the number 1.5 in both 10-adics and 11-adics? And what is -1.5
in both 10-adics and 11-adics? I am going to use 10-adics and 11-adics
because one is a ring and the other a field and use both adic integers
and adic rationals.

As you do not understand the things you are working with at all, and are
not able to do even simple calculations, I will help you. But with what
has been written before you should have been able to do the calculations
yourself ('a' is the digit '10' in the 11-adics):
10-adic 11-adic
1/2 0.5 ...55556
-1/2 ...99999.5 ...55555
3/2 1.5 ...55557
-3/2 ...99998.5 ...55554
and just for fun:
1/11 ...09091 0.1
-1/11 ...90909 ...aaaaa.a

See a pattern?

If there is a pattern, none that I wanted.


However, I think I have made progress on the equator line composed of
roots of negative numbers in adics. We know that sqrt -1 for Reals acts
as a orthogonality or 90 degree angle.

That is only a visualisation of the complex numbers, nothing more.

So I lack the poles and need to think about it some more.

The thinking part is right on the dot. But before thinking comes
understanding what you are playing with, and you are still lacking that.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Thanks Dik, I think persistance will make up for lack of other
abilities.

Dik, I want to define the Great-Circles in Adics and looking at
10-adics of the numbers ending in zeroes such as ....0000001 and ending
in 9s such as ....999999 so I have 1 and -1. Now, can I combine all the
strings ending in zeroes with all the strings ending in 9s and call
that combining a single Great Circle? Where the end result is
....., ....999997, .....9998, .....99999, ....00000, .....00001,
......000002, ....0000003, ...

Dik can you fill in this table, where only Integers are allowed, no
adic-rationals:

10-adic integers 11-adic integers

1/2 1.3
-1/2 -1.3
1.6 2.75
-1.6 -2.75

Dik, what I was hoping for in the defining of Great Circles was that I
could combine infinite strings to other strings such as ...7827821,
......7827822, ....7827823, ... and a different string such as say
3593599, ....3593598, ....3593597,... In such a fashion that combining
the two yields a new Great Circle as this:

...., .... 3593599, ....3593598, ....3593597, ...7827821, .....7827822,
.....7827823, ...

Dik, if this makes a Great Circle---
....., ....999997, .....9998, .....99999, ....00000, .....00001,
......000002, ....0000003, ...

then what do I have to do to make an infinite number of Great Circles
in a given Adic Integer?

As for the poles, in one of your replies, you mentioned that as the n
value of an N-adic increases that the idempotents diverge or separate
in distance. By diverging I mean that in 10-adics the idempotents start
with the digit 6 and digit 5, which is a divergence of 1. So in
20-adics is the idempotent diverging by say 5 where the first digit is
10 and 5? I cannot have poles if they are separated by a distance of
merely one unit, rather they have to be half away apart on a great
circle.

Dik, do the idempotents become somewhat half way apart in distance as
you increase the n value of an N-adic?

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

wrote:
Dik T. Winter wrote:
In article .com writes:
Now what are the other great circles in 10-adics? From what I
understand from Dik, the 10-adics integers is a ring and so would have
additive inverses. So what is the number 1/2 in 10-adics and in
11-adics? What is the number -1/2 in both 10-adics and 11-adics? Then
what is the number 1.5 in both 10-adics and 11-adics? And what is -1.5
in both 10-adics and 11-adics? I am going to use 10-adics and 11-adics
because one is a ring and the other a field and use both adic integers
and adic rationals.

As you do not understand the things you are working with at all, and are
not able to do even simple calculations, I will help you. But with what
has been written before you should have been able to do the calculations
yourself ('a' is the digit '10' in the 11-adics):
10-adic 11-adic
1/2 0.5 ...55556
-1/2 ...99999.5 ...55555
3/2 1.5 ...55557
-3/2 ...99998.5 ...55554
and just for fun:
1/11 ...09091 0.1
-1/11 ...90909 ...aaaaa.a

See a pattern?

If there is a pattern, none that I wanted.

Dik can you fill in this table, where only Integers are allowed, no
adic-rationals:

10-adic integers 11-adic integers

1/2 1.3
-1/2 -1.3
1.6 2.75
-1.6 -2.75

Yes, he can, and I can. But you should do it yourself, otherwise,
you'll only come back with questions similar to this one. These
computations are possible because the n-adics are more than "every
possible combination of the digits 0,1,2,...,n-1"; there's actual
algebraic structure here, which you seem not to care about.

For 1/2: Can you find a 10-adic N such that 2*N = 1 = ...00001_10? No,
since the last digit of 2*N is even, and 1 is odd.

For -1/2: Since 1/2 is not a 10-adic, its additive inverse isn't,
either.

For 1.6: 1.6 = 16/10 = 8/5. Can you find a 10-adic N such that 5*N = 8?

For 1.3 (as an 11-adic): 1.3 = 13/10; can you find an 11-adic N such
that
10*N = 13? (In base 11, this will look like a_11*N = 12_11.) If d is
the last digit of N, then
10 * d = 13 (mod 11) = 2 (mod 11) (numbers in base 10). You have to
find a number between 0 and 10 such that the remainder of 10*d divided
by 11 is 2. There will be exactly one, namely d=9.

What's the digit right before the 9? Call it e. Then you need to have

(11 * e + 9) * 10 = 13 (mod 11^2), which means
110 e + 90 = 13 (mod 121), or
110 e = -77 = 44 (mod 121). Now, let's look at the possibilities:

110 * 0 = 0, which has a remainder of 0 when divided by 121.
110 * 1 = 110, which has a remainder of 110 when divided by 121.
110 * 2 = 220, which has a remainder of 99 when divided by 121.
110 * 3 = 330, which has a remainder of 88 when divided by 121.
(See the pattern? If not, continue.)
110 * 7 = 770, which has a remainder of 44 when divided by 121.

So e = 7. So 1.3 as an 11-adic ends in 79. Continue ad nauseum.

[...] As for the poles, in one of your replies, you mentioned that as the n
value of an N-adic increases that the idempotents diverge or separate
in distance. By diverging I mean that in 10-adics the idempotents start
with the digit 6 and digit 5, which is a divergence of 1.

and also with 0 and 1.

So in
20-adics is the idempotent diverging by say 5 where the first digit is
10 and 5? [...]

The last digit (d) of a 20-adic which is an idempotent must satisfy the
equation
d*d = d (mod 20).

Does 10 satisfy this equation? Does 5?

--- Christopher Heckman

Nov 2, A.P wrote:
My question is, as the n value increases in the n-adic, do the
idempotents spread apart so to speak from say 6... and 5.... in
10-adics do they spread apart such as 7... and 4.... or is there no
pattern of noteworthiness.

Dik Winter wrote:
A pair of idempotents has last digits k and n + 1 - k. You can not
say much otherwise.

A.P. writes:

Okay, in 60-adics can we have idempotents of starting with first digit
of say 15 and 46 where the separation in distance is somewhat equal of
30 units apart?

I need idempotents as poles of a sphere and being poles they have to be
halfway apart in distance. They cannot be next to one another such as 0
and 1 in Reals where they are a unit distance apart.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

In article .com writes:
....
Okay, in 60-adics can we have idempotents of starting with first digit
of say 15 and 46 where the separation in distance is somewhat equal of
30 units apart?

Are there idempotents in the 60-adics with starting digit 15 or 46?
For your information, the starting digits of the idempotents in the
60-adics (except the numbers 0 and 1) are 16, 21, 25, 36, 40 and 45
(yes, there are six).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

In article . com writes:
....
As for the poles, in one of your replies, you mentioned that as the n
value of an N-adic increases that the idempotents diverge or separate
in distance.

That is not what I said. What I said was that if k is an idempotent in
the n-adics, that n - k + 1 is also an idempotent. They can however be
one apart. In the n-adics with n contains two distinct primes only (so
that there are 2 idempotents distinct from 0 and 1), the last digits are
1 apart for the following n: 6, 10, 14, 18, 22, 26... Do you see a
pattern? (Actually you can say a bit more about k: k sqrt(n).)
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Dik T. Winter wrote:
In article .com writes:
...
Okay, in 60-adics can we have idempotents of starting with first digit
of say 15 and 46 where the separation in distance is somewhat equal of
30 units apart?

Are there idempotents in the 60-adics with starting digit 15 or 46?
For your information, the starting digits of the idempotents in the
60-adics (except the numbers 0 and 1) are 16, 21, 25, 36, 40 and 45
(yes, there are six).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Dik, if I define the first Great-Circle in 10-adics as the joining of
the *ending in zeroes* string with the *ending in 9s* string:

......, .... 999997, ....99998, .....99999, ...00000, ....00001,
.....000002, ......

obviously I need more Great Circles than one of them, in fact I need an
infinite supply of great-circles. That is why I asked what the
repetition of 1/2,-1/2, what the repetition of 1.6, -1.6, what the
repetition of 2.75,-2.75 in order to get repeating strings to attach
together as a great-circle.

But the great circles form longitude lines and I need to find the
equator line of a great circle.

Dik, so I wonder about this great circle line:

......, .... 999997, ....99998, .....99999, ...00000, ....00001,
.....000002, ......

An equator would intersect in 2 and only 2 points of a longitude. And
intersect at 90 degrees in those two points. Question Dik or Chris, in
that line defined as a great circle, is it true that there are 2 and
only 2 roots of a negative number to be found in that 10-adics?

Is there a root of a negative number that ends in 9s? Is there a root
of a negative number that ends in 0s? And are those unique roots so
that they are 2 and only 2 roots of negative numbers?

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies