Tjyylli

Consider 2 arbitrary, fixed points A and B on the plane. Suppose you can move from point A in unit distance at any angle to another point and from this point you can again travel a unit distance at any angle to another point and so on. Ultimately your goal is to travel from point A to point B along a path of unit-distance line segments without repeating a point during your journey. Is the set of such paths countable or uncountable?

I believe it is uncountable and here is my thought process, but I'm not sure of my logic. Consider the line between the two points equidistant from them; call this line L. A path of only unit distances can be made between A and any point, P, which lies on L without crossing L(I don't know how to prove this statement but it seems true). This path can be mirrored on the other side of L to connect B to P. Thus for any point, P, which lies on L a path can be made from A to B crossing through P halfway through the path. Since the points on L are uncountable the set of paths between A and B are also uncountable.

Yes, this looks convincing. For the missing step, go directly from A towards P in unit steps until the distance left is less than 2. Then use the remaining distance as the base of an isosceles triangle with unit legs, which you make point away from L.

Go one unit from $a$ at an angle of $t$ to $c$.

Go in unit steps along ca until one is less than a unit away from $b$ to a point $p$.

If $p neq b$, then draw a triangle with base $pb$ and sides of unit length adding the sides as the final steps.

As for each $t$ in $[0,2pi)$, I've constructed a different accepted zigzaging from $a$ to $b$, there are uncountably many ways of so staggering from $a$ to $b$.

Let $C$ be on the line through $B$ that is perpendicular to the segment $AB$ with the distance $BC$ equal to $1/2.$ Take any half-line $L$, not through $B$, that originates at $A$ and intersects the segment $BC.$ For some $nin Bbb N$ there is a path along $L,$ starting at $A,$ determined by $n$ points $A=A_1,...,A_n$ where $A_j,A_{j+1}$ are distance $1$ apart for each $j<n,$ and such that the distance from $A_n$ to $B$ is less than $1.$

Let the point $D$ be such that $A_nD=BD=1$ nd $Dnot in {A_1,...,A_n}.$ Then the path determined by ${A_1,...,A_n}cup {D}$ is a path of the desired type.

The cardinal of the set all such $L$ is $2^{aleph_0}$ so there at least this many paths of the desired type, joining pairs of points .

And each path is determined by a function from some ${1,2,...,m}subset Bbb N$ into $Bbb R^2.$ The set of all such functions has cardinal $2^{aleph_0}$ so there are at most $2^{aleph_0}$ paths of the desired type.

Suppose first that the distance $AB$ between $A$ and $B$ is strictly between $2$ and $3$. Then for every point $C$ such that $AC=1$ and $BC<2$, there is a second point $D$ such that $CD=1=BC$ (draw an isosceles triangle). The set of such points $C$ forms an open arc on the unit circle centered at $A$, and so there are uncountably many such $C$, hence uncountably many such "unit paths".

For general points $A$ and $B$, just find one unit path between $A$ and $B$ that contains two intermediate points $A'$ and $B'$ with $2<A'B'<3$ (for example, points on a giant circle eventually connecting $A$ to $B$), and then vary the path from $A'$ to $B'$ in the way described above.

This proof even shows that one can get uncountably many unit paths with the same number of unit-steps. A slight sharpening shows that for any integer $k>AB$, there are uncountably many unit paths from $A$ to $B$ with exactly $k$ unit-steps.

I believe it is uncountable also, and here is my thought process:

For each direction vector $vinBbb R^2=T_aBbb R^2$, take a curve $alpha_v$ from $a$ to $b$ with $alpha_v'(0)=v$.

Then if $vneq w$, we have $alpha_vneqalpha_w$.

But clearly there are uncountably many direction vectors in $Bbb R^2=T_aBbb R^2$.

Your proof is correct. We can fill in the details on the first part: For any two distinct points $A$ and $B$ in the plane, we can construct a path consisting of unit distances from $A$ to $B$.

If $A$ is more than one unit away from $B$, start at $A$ and take unit steps toward $B$ until we are less than one unit from $B$. If $B$ is a whole number of steps away, we will land on $B$. If not, we will land on a point $P$ within the unit circle centered at $B$. Since $P$ is less than a distance of $1$ away from $B$, the unit circle centered at $P$ will intersect the unit circle centered at $B$ in two places. Take one step from $P$ to an intersection point and then one step from the intersection point to $B$ to finish the path.