Lemma 2.1
Six real records
R = 6
Each opening keeps six real place pages from the live archive in front of the reader.
Mathematical depth 2
Six real places, with a deeper way through.
The six place choices are real pages from the live archive. The mathematical doors underneath keep the journey inside Discover Alone, add one to its depth and open another changing set of six places.
- Start here: Circular WalksVisitor information · North Norfolk
- Try this: Public FootpathVisitor information · North Norfolk
- Try this: Mapped graveyardGraveyard · North Northamptonshire
- Try this: Woodley PlantationWood · North Yorkshire
- Try this: All SaintsGraveyard · Sandwell
- Try this: Walton Maritime Museum, Walton-on-the-Naze, EssexHeritage place · Tendring
The mathematics of finding without an ending
These lemmas describe what the page really does. The formulas change with the carried depth and the live archive total; the facts underneath them do not.
Lemma 2.2
Two useful kinds of choice
B_total = 120 + 6 = 126
The main web combines mathematical continuation with direct entry into real place pages.
Lemma 2.3
One map point
P = (latitude, longitude)
A trustworthy coordinate pair lets one real record appear at its saved position on the map.
Lemma 2.4
The complete archive
N = 1,118,481
N is read from the same live place database that makes the public pages.
Lemma 2.5
Real-page share
6 / 126 = 4.7619%
Six of the main choices open real records, so the archive remains part of every Fractal page.
Lemma 2.6
Distance on the earth
a = sin^2(delta_lat/2) + cos(lat_1) cos(lat_2) sin^2(delta_long/2)
The map uses positions on a globe, so nearby means surface distance rather than a flat drawing.
Lemma 2.7
Wrapped selection
row = ((start + offset - 1) mod N) + 1
When a chosen position passes the end of the database, it continues from the beginning without inventing a record.
Lemma 2.8
Continuation share
120 / 126 = 95.2381%
Most main choices deepen the mathematical web instead of ending the journey after one page.
Lemma 2.9
Four-country whole
N = Ireland + England + Scotland + Wales
The four country parts add back to the complete live archive rather than creating four separate totals.
Lemma 2.10
Far-apart starting offsets
offsets = {1, 97, 997, 10,007, 50,021, 100,003}
The six choices are taken from widely separated positions rather than six neighbouring database rows.
Lemma 2.11
Possible labelled routes
routes after k mathematical steps = 120^k
The formula counts possible sequences of labelled doors; it does not pretend that extra database records exist.
Lemma 2.12
One row, one public address
public page = page(saved record identifier)
A saved database row and its public page remain the same thing; no second made page is required.
Continue through 120 mathematical doors
Every door below remains inside Discover Alone, carries depth 3 and opens six more real place pages.