Subject: Re: halfneighbours<br/>
Date: Apr 22, 2002 @ 15:50<br/>
Author: acroorca2002 (&quot;acroorca2002&quot; &lt;orc@...&gt;)<br/>
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<div id="ygrps-yiv-2029525499">i love all this loopy math<br/>
<br/>
first of all david each brick in a wall might have as many as 28 <br/>
neighbors by the fullest application of your corner reckoning system<br/>
<br/>
& brendan & david<br/>
punctologically speaking<br/>
which i admit is upside down from your view<br/>
the corners of the square have a value not of 1 half<br/>
nor the square root of 2<br/>
but<br/>
since they are dipoints<br/>
they have a value of 2 each<br/>
<br/>
so the entire border of the square has a value of 12<br/>
& this even without regard to its surface area yet<br/>
<br/>
<br/>
the 6 faces of the brick are only of fortuitous relevancy<br/>
rather like the 6 points of a touchdown are like the 6 points of 2 <br/>
field goals <br/>
or the 6 points of 3 safeties for that matter<br/>
<br/>
but since the brick has been thrown in<br/>
it is nice to observe that the borders of each of its 6 faces have a <br/>
value of 12 just like the original square<br/>
& thus the entire brick has a value of 72 when disassembled into its <br/>
6 faces<br/>
& also without regard to its surface area yet<br/>
<br/>
but when the brick is viewed within the scheme presented & as a whole<br/>
meaning as the 3d object that it is<br/>
its 6 faces have a value of 1 each<br/>
& its 12 edges have a value of 2 each<br/>
& its 8 corners have a value of 3 each<br/>
<br/>
so its full value is not 72 any longer but only 56<br/>
<br/>
do how did it cost the brick only 16 points to transfer from the 2nd <br/>
to the 3rd dimension<br/>
<br/>
it spent so little because the surface of the original square was not <br/>
counted as a point<br/>
as it should have been in punctology<br/>
<br/>
by the correct math<br/>
a square has 1 surface of value 1<br/>
4 edges of value 2 each<br/>
& 4 corners of value 3 each<br/>
for a total value of 21<br/>
<br/>
& a brick or any cuboid has a mass of value 1<br/>
6 faces of value 2 each<br/>
12 edges of value 3 each<br/>
& 8 corner points of value 4 each<br/>
for a total value of 81<br/>
<br/>
by this reckoning the brick spends fully 45 points in transferring <br/>
from the 2nd to the 3rd dimension<br/>
which is coincidentally more in keeping with what one expected<br/>
<br/>
m<br/>
<br/>
<blockquote><span title="qreply"> --- In BoundaryPoint@y..., &quot;Brendan Whyte&quot; &lt;b.whyte@p...&gt; wrote:<br/>
&gt; But as the distance from the centre of a white square (say to the <br/>
centre of <br/>
&gt; the neighbouring black squares is 1, and the distaince to the <br/>
neighbouring <br/>
&gt; &#39;half neighbour&#39; white squares is sqr2=1.414...<br/>
&gt; SShould not the &#39;half neighbours&#39; really be (1/sqr2)=0.707 <br/>
neighbours, for <br/>
&gt; 6.828 neighbours total ~7?<br/>
&gt; BW<br/>
&gt; <br/>
&gt; &gt;In geographic information systems back in the 1970s, we used to <br/>
use a term<br/>
&gt; &gt;&quot;half-neighbor&quot;. In a checkerboard, each square has 4 &#39;full <br/>
neighbors&#39; (of<br/>
&gt; &gt;the opposite color to the current square), and four half-neighbors <br/>
(same<br/>
&gt; &gt;color), for a total of six neighbors.  Just as each brick in a <br/>
wall has<br/>
&gt; &gt;six neighbors.<br/>
&gt; <br/>
&gt; <br/>
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