Table view of an n-dimensional address space

[m***@lycos.com]

[My apologies if this post is OT. I have no idea where I should post
this]

I need help in decoding the following excerpt from an email from our
system architect. Can someone kindly explain to me, in layman terms,
what this email states?

...You can implement this in software either way.
After all, an n-dimensional address space, with
labeled axes, is isomorphic with a table with n
labeled input columns and 1 output column
(something like an enhanced truth table)....


Thanks,
Masood

[Pascal Bourguignon]

Post by m***@lycos.com
[My apologies if this post is OT. I have no idea where I should post
this]
I need help in decoding the following excerpt from an email from our
system architect. Can someone kindly explain to me, in layman terms,
what this email states?
...You can implement this in software either way.
After all, an n-dimensional address space, with
labeled axes, is isomorphic with a table with n
labeled input columns and 1 output column
(something like an enhanced truth table)....

Assume you have the 3D table:

#3A(((101 102 103 104) (111 112 113 114) (121 122 123 124))
((201 202 203 204) (211 212 213 214) (221 222 223 224)))

You can print it as:

(let ((table #3A(((101 102 103 104) (111 112 113 114) (121 122 123 124))
((201 202 203 204) (211 212 213 214) (221 222 223 224)))))
(format t "~***@A ~***@A ~***@A : ~4A~%" 'i 'j 'k 'table)
(loop for i from 0 below 2 do
(loop for j from 0 below 3 do
(loop for k from 0 below 4 do
(format t "~3D ~3D ~3D : ~4D~%" i j k (aref table i j k))))))

I J K : TABLE
0 0 0 : 101
0 0 1 : 102
0 0 2 : 103
0 0 3 : 104
0 1 0 : 111
0 1 1 : 112
0 1 2 : 113
0 1 3 : 114
0 2 0 : 121
0 2 1 : 122
0 2 2 : 123
0 2 3 : 124
1 0 0 : 201
1 0 1 : 202
1 0 2 : 203
1 0 3 : 204
1 1 0 : 211
1 1 1 : 212
1 1 2 : 213
1 1 3 : 214
1 2 0 : 221
1 2 1 : 222
1 2 2 : 223
1 2 3 : 224

[Richard Heathfield]

Post by m***@lycos.com
[My apologies if this post is OT. I have no idea where I should post
this]

It certainly fits in comp.programming (where I am reading this), so I've
set followups there.

Post by m***@lycos.com
I need help in decoding the following excerpt from an email from our
system architect. Can someone kindly explain to me, in layman terms,
what this email states?
...You can implement this in software either way.
After all, an n-dimensional address space, with
labeled axes, is isomorphic with a table with n
labeled input columns and 1 output column
(something like an enhanced truth table)....

Imagine a set of points in one-dimensional space. Clearly, they all sit
on a line. Perhaps A is at point 3, B at point 7, and C at point 9. We
can picture them like this:

A B C
+----+----+----+----+----+----+----+----+----+----+----+----+
0 1 2 3 4 5 6 7 8 9 10 11 12

That's n-dimensional space, for n=1.

But we could represent it like this instead:

+---+------+
| X | Data |
+---+------+
| 3 | A |
| 7 | B |
| 9 | C |
+---+------+

which is a bit more compact, obviously.

Now imagine a set of points in two-dimensional space:


5 +
|
|
4 + D
|
|
3 + A E
|
|
2 + C
|
|
1 + B
|
|
0 +----+----+----+----+----+----+----+----+----+----+
0 1 2 3 4 5 6 7 8 9 10

Again, we can represent this information more compactly as follows:

+---+---+------+
| X | Y | Data |
+---+---+------+
| 4 | 3 | A |
| 2 | 1 | B |
| 6 | 2 | C |
| 3 | 4 | D |
| 7 | 3 | E |
+---+---+------+

Now imagine n-dimensional space for n=3. In a Usenet article, I can't
even /draw/ this as a graph. But in table form, it's easy:

+---+---+---+------+
| X | Y | Z | Data |
+---+---+---+------+
| 3 | 2 | 1 | A |
| 1 | 7 | 6 | B |
| 4 | 1 | 1 | C |
| 1 | 8 | 8 | D |
| 5 | 2 | 0 | E |
| 9 | 8 | 3 | F |
| 2 | 1 | 3 | G |
+---+---+---+------+

and indeed this form can be extended arbitrarily into n dimensions
without ever needing more than a two-dimensional table in which to
record the co-ordinate information.

HTH. HAND.