I don't think this can be done on Earth without resorting to infinitesimals:
You walk 1 mile north. This leaves you 1 mile north of your house.
You walk 2 miles east. Your longitude depends on your latitude, but you're still 1 mile north
of your house.
Now if you start walking north and walk straight, you are following a great polar circle which
means you can only go in two directions - due north and due south (this is in contrast to 'start
due east and walk straight', which causes you to move north or south depending on which
hemisphere you're in). So the only way you can walk back to your house in 1 mile is by
walking all that distance due south. Since you start by walking north, you must start an
infinitesimal distance away from the north pole so that you immediately start walking south.
Therefor you live 1 and an infinitesimal bit miles from the North pole. You walk north, travel
2 miles east in an infinitely tight circle, which somehow changes your longitude by 180°, then
walk north for an infinitesimal distance before heading south for the remaining mile.
You're right. I think I'll need to change it a bit. Instead of walking
east for 2 miles, walk east for Pi (3.1415...) miles. Also, to avoid
confusion, I'll change the wording of the first statement a bit.
This new one *can* be done:
Leave you house and start walking north. After walking in a straight
line for a mile, turn and walk east for Pi miles. Then turn left again
and stark walking north. Keep walking straight for 1 mile and you'll be
back at your house.
Ok. This one works.
--Rein
I'm trying to remember my logic here. Work is distracting! Actually,
there are any number of possibilities, from walk east 1/x miles (where
X is any whole number) to walk east approximately 2XPi miles. It
doesn't have to be walk east Pi miles, I just thought it would confuse
people.
And the turn left "again" IS a hint.
Any one know how this works?
--Rein
You live (1 - 1/2n) miles from the North Pole, eg 1/2 mile, 3/4 mile, 5/6 mile away from the
pole. The first mile takes you 1/2n miles away from the North Pole. You then walk east,
making n complete revolutions, so that doesn't take you anywhere. You then turn to face
North and go back the way you came.
Well, we'll consider the area within 5 miles of the North Pole as flat
enough! it's only off by a factor of + 10^-3 miles (I think), so you'll
still find your house whether you calculate it with the curvature of the
earth in consideration or not.
But, consider starting you start 1/2 miles south of the north pole. You
walk right over the north pole, but you keep going straigh, so you walk
another 1/2 mile SOUTH and stop 180 degrees from where you were,
still at the same latitude. In this case, considering the top of the earth
to be flat, you ar at a latitude with length <pi> (2<pi>r = C, r = 1/2,
so C = <pi>😉. So you walk <pi> miles east, and yes, you are back
where you started (very close on a curved earth) so you are right and I
just lost my concentration earlier and am therefor wrong. (My poor
head). If you start at a point 1 - 1/x miles from the north pole, where
x is any whole number, it will work. The key is that you end up at a
location 1/x miles from the north pole. I flipped the circumference and
radius around in my head.
Long story short: you are right, I am confused. Congrats. If you want
to figure out how to account for the curvature of the Earth, be my
guest. I'm not going to worry about it.
--Rein
Main Entry: [1]line
Pronunciation: 'lIn
Function: noun
Usage: often attributive
Etymology: Middle English; partly from Old French ligne, from Latin
linea, from feminine of lineus made of flax, from linum flax; partly
from Old English lIne; akin to Old English lIn flax &#8212;more at LINEN
Date: before 12th century
1 a : THREAD, STRING, CORD, ROPE: as (1) : a comparatively strong
slender cord (2) : CLOTHESLINE (3) : a rope used on shipboard b (1) :
a device for catching fish consisting of a cord with hooks and other
fishing gear (2) : scope for activity : ROPE c : a length of material
used in measuring and leveling d : piping for conveying a fluid (as
steam) e (1) : a wire or pair of wires connecting one telegraph or
telephone station with another or a whole system of such wires; also :
any circuit in an electronic communication system (2) : a telephone
connection <tried to get a line>; also : an individual telephone
extension <a call on line 2> (3) : the principal circuits of an electric
power system
2 a (1) : a horizontal row of written or printed characters; also : a
blank row in lieu of such characters (2) : a unit in the rhythmic
structure of verse formed by the grouping of a number of the smallest
units of the rhythm (as metrical feet) (3) : an often numbered section
of a computer program containing a single command or a small
number of commands b : a short letter : NOTE c : plural : a certificate
of marriage d : the words making up a part in a drama &#8212; usually used
in plural e : any of the successive horizontal rows of picture elements
on the screen of a cathode-ray tube (as a television screen)
3 a : something (as a ridge or seam) that is distinct, elongated, and
narrow b : a narrow crease (as on the face) : WRINKLE c : the course
or direction of something in motion : ROUTE d (1) : a state of
agreement or conformity : ACCORDANCE (2) : a state of order,
control, or obedience <you're getting out of line> e : a boundary of an
area <the state line> f : the track and roadbed of a railway g : an
amount of cocaine that is arranged in a line to be inhaled through the
nose
4 a : a course of conduct, action, or thought b : a field of activity or
interest c : a glib often persuasive way of talking
5 a : LIMIT, RESTRAINT b : archaic : position in life : LOT
6 a (1) : FAMILY, LINEAGE (2) : a strain produced and maintained
especially by selective breeding or biological culture (3) : a
chronological series b : dispositions made to cover extended military
positions and presenting a front to the enemy &#8212; usually used in plural
c : a military formation in which the different elements are abreast of
each other d : naval ships arranged in a regular order e (1) : the
combatant forces of an army distinguished from the staff corps and
supply services (2) : the force of a regular navy f (1) : officers of the
navy eligible for command at sea distinguished from officers of the
staff (2) : officers of the army belonging to a combatant branch g : an
arrangement or placement of persons or objects of one kind in an
orderly series <a line of trees> <stand on line> <waiting in line>;
also : the persons or objects so positioned <the line moved slowly at
the bank> h (1) : a group of public conveyances plying regularly under
one management over a route (2) : a system of transportation
together with its equipment, routes, and appurtenances; also : the
company owning or operating it i : a succession of musical notes
especially considered in melodic phrases j (1) : an arrangement of
operations in manufacturing permitting sequential occurrence on
various stages of production (2) : the personnel of an organization
that are responsible for its stated objective k (1) : the 7 players
including center, 2 guards, 2 tackles, and 2 ends who in offensive
football play line up on or within one foot of the line of scrimmage
(2) : the players who in defensive play line up within one yard of the
line of scrimmage
7 : a narrow elongated mark drawn or projected: as a (1) : a circle of
latitude or longitude on a map (2) : EQUATOR b : a mark (as on a
map) recording a boundary, division, or contour c : any of the
horizontal parallel strokes on a music staff on or between which notes
are placed &#8212; compare SPACE d : a mark (as by pencil) that forms part
of the formal design of a picture distinguished from the shading or
color e : a division on a bridge score dividing the score for bonuses
from that for tricks f (1) : a demarcation of a limit with reference to
which the playing of some game or sport is regulated; usually used in
combination (2) : a marked or imaginary line across a playing area
(as a football field) parallel to the end line (3) : LINE OF SCRIMMAGE
8 : a straight or curved geometric element that is generated by a
moving point and that has extension only along the path of the
point : CURVE
9 a : a defining outline : CONTOUR b : a general plan : MODEL &#8212;
usually used in plural
10 a : chiefly British : PICA; used to indicate the size of large type b :
the unit of fineness of halftones expressed as the number of screen
lines to the linear inch
11 : merchandise or services of the same general class for sale or
regularly available
12 a : a source of information : INSIGHT b : betting odds offered by a
bookmaker especially on a sporting event
13 : a complete game of 10 frames in bowling &#8212; called also string
14 : LINE DRIVE
- liny also lin.ey adjective
- between the lines 1 : by implication : in an indirect way 2 : by way of
inference
- down the line : all the way : FULLY
- in line for : due or in a position to receive
- on line : in or into operation
- on the line 1 : in complete commitment and at great risk <puts his
future on the line by backing that policy> 2 : on the border between
two categories 3 : IMMEDIATELY <paid cash on the line>
Pronunciation Key
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Did you actually expect me to read that?
Ok, so no, they're not geometrically straight lines. However, I think
the inaccuracy is easier to put up with than saying "walk in a not quite
straight considering the curvature of the earth line for 1 mile" every
time. Also, I'm not taking the curvature of the earth into consideration
when I'm calculating the circumferences of the lines of latitude.
However, this close to the earth, I'm only off by a factor of .001, i.e.
plus or minus decimeters on the kilometer, so I'm not going to worry
about it.