Pyramid Prospect Proposal
This is a study and proposal to use a Diamond patten rather than the traditional Square patten for prospecting limestone blocks for our larger pyramids.
The traditional method.
| Normally we've been following a square patten when looking for blocks, with the build site in the center. This problem with this is when pushing, we can only move the blocks in the cardinal directions, and that means that blocks in the corners need to be pushed twice as far as blocks at the center of the edges. Consider the site to the right. A block at the upper-left corner needs to be pushed right to the center of the edge, and then down to the build site - a distance of 158 coordinates. A block that was already at the center of the top row only needs to be pushed 79 coordinates. (The blue square at the center is one-half of the area prospected, and is common to all figures in this analysis.) |  |
The proposed method.
| I would suggest that a diamond shape would be more efficient. In the figure to the right, the area covered by the green diamond is the same as the red square in the method above, and so should contain the same number of blocks. The difference is that a block at one of the corners only needs to be pushed 112 coordinates. In fact, any block on the outside edge needs to be moved the same 112 distance. |  |
The difference.
| Now let's look at the actual difference this makes. Figure C shows the overlay of the two methods. Any block in the purple octagon would be in both pattens, and so would not be effected. The difference is that the blocks that would have traditionally been in the red areas are now in the green areas, and require less pushing to move to the center of the site. |  |
An analysis
Okay, that seems to make sense, but just how much of a difference is this going to make ?We are currently working on a 1240 block pyramid in CotS that I'm using to do a rough estimate as to the savings of following the diamond method. We have tapped about 9,000 rods, and have about 450 blocks located. So that gives us one block every 20 rods.
So to crack 1/2 of the blocks (620) we need an area of 12,400 coordinates square (about 112x112) - the blue area in the figures above. For all blocks, we need an area of 24,800 coordinates square (about 158x158 square - the red area, or a diamond with a height/width of 224 - the green area).
The areas of difference (the red/green triangles in Fig C) cover a combined area of 4,356 coordinates, and should count for about 218 blocks.
The worst case in the red areas would be at a corner, and that block would need to be pushed 158 coordinates. In the green area, the worst case is 112 coords, a difference of 46 coords. Blocks at the points where the green and red areas meet would have to be pushed the same distance. So let's assume the average distance saved is 46 / 2, or 23 coordinates.
Update : I calculated the average coords to move all possible blocks in a Red and Green triangle. The actual average saving was 26 coords, better than the above guess. That woulds out to an average saving of 4.5 coords for all blocks. I verified this by calculating the average coords to move every possible block in the Traditional and Diamond pattens, and again came up with a difference of 4.5 -- Siara
Multiplying that by the 218 blocks in these areas give us a total savings of 26*218 = 5,668 coordinates that we don't have to push.
If we average that over all of the blocks in the site, 5,668 / 1,240 = 4.5 coords.
Conclusion
By following this new patten, we effectively have a savings equivalent of moving every block in the site 4.5 coordinates closer to the build site - which I feel is a very significant difference.Expressed as time saved : 26*218=5,668 coords - 8 pushes to a coord 8*5,668 = 45,344 pushes - if we can push once per second = 12 person hours of pushing eliminated. That's about a 5.7% savings over all.
Search process
In order to make our search site fit the diamond patten, we would search in the usual square method until we have cracked 1/2 of the total blocks (the blue square). Then, when setting new rows of rods, simply skip the first coordinate in the rows at both ends. If that does not give us the total blocks needed, the diamond can be extended anywhere, as any point on it's edge is the same distance to the center.
Savings for each Pyramid type
| Teirs | Blocks | Area | Square Length | Ave Coord | Diamond Width | Ave Coord | Ave. Save. | Total Save. | Time (Days) H:M |
|
| People | 3 | 14 | 280 | 17 | 8.5 | 24 | 8.0 | 0.5 | 6.8 | 00:00 |
| Fertile Land | 5 | 55 | 1100 | 34 | 17.0 | 48 | 16.0 | 1.0 | 53.5 | 00:07 |
| Deep Ocean | 7 | 140 | 2800 | 53 | 26.5 | 75 | 25.0 | 1.5 | 212.2 | 00:28 |
| Renewal | 11 | 506 | 10120 | 101 | 50.5 | 143 | 47.6 | 2.9 | 1461.4 | 03:14 |
| Distant Plains | 15 | 1240 | 24800 | 158 | 79.0 | 223 | 74.5 | 4.5 | 5602.4 | 12:26 |
| Many Wonders | 19 | 2470 | 49400 | 223 | 111.5 | 315 | 105.1 | 6.4 | 15750.7 | 1 11:00 |
| <Unknown> | 25 | 5525 | 110500 | 333 | 166.5 | 471 | 157.0 | 9.5 | 52610.7 | 4 20:54 |
This presumes there is one block per 20 coords, and Time presumes we get one push per second, and there are 8 pushed to move one coord.
