On the magic of some familiesof fractal dendrites

Ligia Loretta Cristea
FWF stand-alone project P27050-N26
Universität Graz
joint work with Bertran Steinsky and Gunther Leobacher

Numeration 2018, Paris, May 22-25, 2018

• LLC, B. Steinsky, Paths of infinite length in 4 × 4 -labyrinth fractals, Geometriae Dedicata (2009)• LLC, B. Steinsky, Curves of Infinite Length in Labyrinth

Fractals, Proceedings of the Edinburgh Math. Soc. (2010)• LLC, B. Steinsky, Mixed labyrinth fractals, J. Topology andits Applications (2017)• LLC, G. Leobacher, A note on lengths of arcs in mixed

labyrinth fractals, Monatshefte f. Mathematik (2017)• LLC, G. Leobacher, Supermixed labyrinth fractals,

submitted 2018

What is a labyrinth fractal?

• 4 ”doors”: golden, very ”small”• from any of the ”doors” a path is starting• all ”doors” are connected• infinitely many, infinitely small rooms:• only one path between any two (rooms)• this unique path is infinitely long (under certain conditions)What is a labyrinth fractal?

• 4 ”doors”: golden, very ”small”• from any of the ”doors” a path is starting• all ”doors” are connected• infinitely many, infinitely small rooms:• only one path between any two (rooms)• this unique path is infinitely long (under certain conditions)What is a labyrinth fractal?

• 4 ”doors”: golden, very ”small”• from any of the ”doors” a path is starting• all ”doors” are connected• infinitely many, infinitely small rooms:• only one path between any two (rooms)• this unique path is infinitely long (under certain conditions)What is a labyrinth fractal?

• 4 ”doors”: golden, very ”small”• from any of the ”doors” a path is starting• all ”doors” are connected• infinitely many, infinitely small rooms:• only one path between any two (rooms)• this unique path is infinitely long (under certain conditions)What is a labyrinth fractal?

• 4 ”doors”: golden, very ”small”• from any of the ”doors” a path is starting• all ”doors” are connected• infinitely many, infinitely small rooms:• only one path between any two (rooms)• this unique path is infinitely long (under certain conditions)What is a labyrinth fractal?

• 4 ”doors”: golden, very ”small”• from any of the ”doors” a path is starting• all ”doors” are connected• infinitely many, infinitely small rooms:• only one path between any two (rooms)• this unique path is infinitely long (under certain conditions)What is a labyrinth fractal?

• 4 ”doors”: golden, very ”small”• from any of the ”doors” a path is starting• all ”doors” are connected• infinitely many, infinitely small rooms:• only one path between any two (rooms)• this unique path is infinitely long (under certain conditions)(Labyrinth) patterns. The graph of a (labyrinth) pattern

A 4 × 4 (labyrinth) pattern and its graph

What is a labyrinth pattern?

Property 1 (The Tree Property)

The graph of the labyrinth pattern

Property
The graph of the labyrinth pattern is a tree. (the Tree Property)

Property 2 (The Exits Property)
Property
There is exactly one horizontal and exactly one vertical exit pair inthe labyrinth pattern. (the Exits Property)

Property 2 (The Exits Property)
Property
There is exactly one horizontal and exactly one vertical exit pair inthe labyrinth pattern. (the Exits Property)

Property 2 (The Exits Property)
Property
There is exactly one horizontal and exactly one vertical exit pair inthe labyrinth pattern. (the Exits Property)

Property 3 (The Corner Property)

Property
If there is a white square at a corner of the labyrinth pattern, thenthere is no white square at the diagonally opposite corner of thelabyrinth pattern. (the Corner Property)

Property 3 (The Corner Property)

Property
If there is a white square at a corner of the labyrinth pattern, thenthere is no white square at the diagonally opposite corner of thelabyrinth pattern. (the Corner Property)

Construction of a labyrinth fractal
A 4 × 4-labyrinth pattern/set.

. . . labyrinth fractal

Construction of a labyrinth fractal
A 4 × 4-labyrinth pattern/set.

. . . labyrinth fractal

Dendrites
Theorem
For all m × m labyrinth patterns, the constructed self-similar
fractal L is a dendrite.

DendriteA dendrite is a connected and locally connected compact Hausdorff
space that contains no simple closed curve.

A Fourth Property

Horizontally Blocked
A labyrinth pattern is called horizontally blocked if the row (ofsquares) from the left exit to the right exit contains at least one
black square.

Vertically Blocked
A labyrinth pattern is called vertically blocked if the column (ofsquares) from the top exit to the bottom exit contains at least one
black square.

A Fourth Property

Horizontally Blocked
A labyrinth pattern is called horizontally blocked if the row (ofsquares) from the left exit to the right exit contains at least one
black square.

Vertically Blocked
A labyrinth pattern is called vertically blocked if the column (ofsquares) from the top exit to the bottom exit contains at least one
black square.

A Fourth Property

Horizontally Blocked
A labyrinth pattern is called horizontally blocked if the row (ofsquares) from the left exit to the right exit contains at least one
black square.

Vertically Blocked
A labyrinth pattern is called vertically blocked if the column (ofsquares) from the top exit to the bottom exit contains at least one
black square.

Self-similar labyrinth fractals. Main Result......

Theorem
Let L∞ be the (self-similar) labyrinth fractal generated by ahorizontally and vertically blocked m × m-labyrinth pattern.
(a) Between any two points in L∞ there is a unique arc a.
log r(b) The length of a is infinite and dimB (a) = logm.

(c) The set of all points, at which no tangent to a exists, is dense
in a.

Self-similar labyrinth fractals. Main Result......

Theorem
Let L∞ be the (self-similar) labyrinth fractal generated by ahorizontally and vertically blocked m × m-labyrinth pattern.
(a) Between any two points in L∞ there is a unique arc a.
log r(b) The length of a is infinite and dimB (a) = logm.

(c) The set of all points, at which no tangent to a exists, is dense
in a.

Self-similar labyrinth fractals. Main Result......

Theorem
Let L∞ be the (self-similar) labyrinth fractal generated by ahorizontally and vertically blocked m × m-labyrinth pattern.
(a) Between any two points in L∞ there is a unique arc a.
log r(b) The length of a is infinite and dimB (a) = logm.

(c) The set of all points, at which no tangent to a exists, is dense
in a.

Self-similar labyrinth fractals. Main Result......

Theorem
Let L∞ be the (self-similar) labyrinth fractal generated by ahorizontally and vertically blocked m × m-labyrinth pattern.
(a) Between any two points in L∞ there is a unique arc a.
log r(b) The length of a is infinite and dimB (a) = logm.

(c) The set of all points, at which no tangent to a exists, is dense
in a.

Self-similar labyrinth fractals

Famous Theorems used for the proofs• Jordan Curve Theorem• Hahn-Mazurkiewicz-Sierpiński Theorem• Perron-Frobenius Theorem• a labyrinth version of the Steinhaus Chessboard Theorem

Mixed labyrinth fractals

Figure: Two labyrinth patterns, A1 (a 4-pattern) and A2 (a 5-pattern)

Figure: The mixed (labyrinth) set W2 , constructed based on the above
patterns A1 and A2 , that can also be viewed as a 20-pattern

Figure: Labyrinth patterns: A1 , A2 (as before), and A3 (4 × 4)

Figure: The mixed (labyrinth) set of level 4 defined by a sequence{Ak }k≥1 where the first three patterns are A1 , A2 , A3 , respectively,
shown above, and the fourth is A1

Topological properties of mixed labyrinth fractals
Lemma
Let {Ak }∞k=1 be a sequence of non-empty patterns, mk ≥ 3, andn ≥ 1. If A1 , . . . An are labyrinth patterns, then Wn is anm(n) × m(n)-labyrinth set (i.e., it has the Tree Property, Q
Exits
Property, Corner property ), for all n ≥ 1, where m(n) = nk=1 mk .
We call the limit set L∞ =\[W the mixed labyrinthn≥1 W ∈Wn

fractal generated by {Ak }∞k=1 .

TheoremLet {Ak }∞k=1 be a sequence of labyrinth patterns, mk ≥ 3, for all
k ≥ 1. Then L∞ is a dendrite.

Topological properties of mixed labyrinth fractals
Lemma
Let {Ak }∞k=1 be a sequence of non-empty patterns, mk ≥ 3, andn ≥ 1. If A1 , . . . An are labyrinth patterns, then Wn is anm(n) × m(n)-labyrinth set (i.e., it has the Tree Property, Q
Exits
Property, Corner property ), for all n ≥ 1, where m(n) = nk=1 mk .
We call the limit set L∞ =\[W the mixed labyrinthn≥1 W ∈Wn

fractal generated by {Ak }∞k=1 .

TheoremLet {Ak }∞k=1 be a sequence of labyrinth patterns, mk ≥ 3, for all
k ≥ 1. Then L∞ is a dendrite.

Topological properties of mixed labyrinth fractals
Lemma
Let {Ak }∞k=1 be a sequence of non-empty patterns, mk ≥ 3, andn ≥ 1. If A1 , . . . An are labyrinth patterns, then Wn is anm(n) × m(n)-labyrinth set (i.e., it has the Tree Property, Q
Exits
Property, Corner property ), for all n ≥ 1, where m(n) = nk=1 mk .
We call the limit set L∞ =\[W the mixed labyrinthn≥1 W ∈Wn

fractal generated by {Ak }∞k=1 .

TheoremLet {Ak }∞k=1 be a sequence of labyrinth patterns, mk ≥ 3, for all
k ≥ 1. Then L∞ is a dendrite.

The construction of the path between exits
Example: The path between the bottom exit and the right exit

Figure: The set W2 constructed with the patterns A1 and A2 shown
before, and the path from the bottom exit to the right exit of W2 (inlighter gray).

One can check that (2) = 48.
The idea of the construction described in the following works for
all mixed labyrinth fractals.

Paths in mixed labyrinth sets. Paths in patterns

Figure: The path from the bottom exit to the right exit of A1• first, we find the path between the bottom and the right exitof W1
• then we denote each white square in the path according to its

neighbours within the path: there are 6 possible types ofsquares: , , , , , and -square

Paths in mixed labyrinth sets. Paths in patterns

Figure: The path from the bottom exit to the right exit of A1• first, we find the path between the bottom and the right exitof W1
• then we denote each white square in the path according to its

neighbours within the path: there are 6 possible types ofsquares: , , , , , and -square

Paths in mixed labyrinth sets. Paths in patterns
In order to obtain the -path in G(W2 ), we replace each -squareof the path in G(W1 ) with the -path in G(A2 ).
Analogously, we do this for the other marked white squares.

Figure: Paths from bottom to top and from left to right exit of A2

Paths in mixed labyrinth sets

In general, for any pair of exits and n ≥ 1, we replace each marked
white square in the path of G(Wn ) by its corresponding path inG(An+1 ) and obtain the path of G(Wn+1 ).

Paths in mixed labyrinth sets
Let {Ak }k≥1 be a sequence of labyrinth patterns, that defines thesequence {Wn }n≥1 of mixed labyrinth sets.

Proposition
There exist non-negative 6 × 6-matrices Mk , k = 1, 2, . . . , suchthat for all n ≥ 1, and for M(n) = M1 · M2 · . . . · Mn , the element
in row x and column y of M(n) is the number of y -squares in thex-path in G(Wn ), for x, y ∈ { , , , , , }.
Furthermore,


 
(n)
1
 (n) 
 1 


 
 (n)  
 = M(n) ·  1  (n) 
 1 


 
 (n) 
 1 
(n)

1

Paths in mixed labyrinth sets
Let {Ak }k≥1 be a sequence of labyrinth patterns, that defines thesequence {Wn }n≥1 of mixed labyrinth sets.

Proposition
There exist non-negative 6 × 6-matrices Mk , k = 1, 2, . . . , suchthat for all n ≥ 1, and for M(n) = M1 · M2 · . . . · Mn , the element
in row x and column y of M(n) is the number of y -squares in thex-path in G(Wn ), for x, y ∈ { , , , , , }.
Furthermore,


 
(n)
1
 (n) 
 1 


 
 (n)  
 = M(n) ·  1  (n) 
 1 


 
 (n) 
 1 
(n)

1

Paths in mixed labyrinth sets
Sketch of the proof
For k ≥ 1, we define the matrix Mk (the path matrix of Ak ):• the columns of Mk from left to right and the rows of Mk from

top to bottom correspond to , , , , , and , (ordered setof indices)• the element in row x and column y of Mk is the number ofy -squares in the x-path in G(Ak ).
One can easily check that the matrix multiplication reflects thesubstitution of paths.
(Proof by induction)
Remark. The proposition yields• in the self-similar case M(n) = M n• in the mixed case M(n + 1) = M(n) · Mn+1

Paths in mixed labyrinth sets
Sketch of the proof
For k ≥ 1, we define the matrix Mk (the path matrix of Ak ):• the columns of Mk from left to right and the rows of Mk from

top to bottom correspond to , , , , , and , (ordered setof indices)• the element in row x and column y of Mk is the number ofy -squares in the x-path in G(Ak ).
One can easily check that the matrix multiplication reflects thesubstitution of paths.
(Proof by induction)
Remark. The proposition yields• in the self-similar case M(n) = M n• in the mixed case M(n + 1) = M(n) · Mn+1

Paths in mixed labyrinth sets
Sketch of the proof
For k ≥ 1, we define the matrix Mk (the path matrix of Ak ):• the columns of Mk from left to right and the rows of Mk from

top to bottom correspond to , , , , , and , (ordered setof indices)• the element in row x and column y of Mk is the number ofy -squares in the x-path in G(Ak ).
One can easily check that the matrix multiplication reflects thesubstitution of paths.
(Proof by induction)
Remark. The proposition yields• in the self-similar case M(n) = M n• in the mixed case M(n + 1) = M(n) · Mn+1

Paths in mixed labyrinth sets
Sketch of the proof
For k ≥ 1, we define the matrix Mk (the path matrix of Ak ):• the columns of Mk from left to right and the rows of Mk from

top to bottom correspond to , , , , , and , (ordered setof indices)• the element in row x and column y of Mk is the number ofy -squares in the x-path in G(Ak ).
One can easily check that the matrix multiplication reflects thesubstitution of paths.
(Proof by induction)
Remark. The proposition yields• in the self-similar case M(n) = M n• in the mixed case M(n + 1) = M(n) · Mn+1

Paths in mixed labyrinth sets
Sketch of the proof
For k ≥ 1, we define the matrix Mk (the path matrix of Ak ):• the columns of Mk from left to right and the rows of Mk from

top to bottom correspond to , , , , , and , (ordered setof indices)• the element in row x and column y of Mk is the number ofy -squares in the x-path in G(Ak ).
One can easily check that the matrix multiplication reflects thesubstitution of paths.
(Proof by induction)
Remark. The proposition yields• in the self-similar case M(n) = M n• in the mixed case M(n + 1) = M(n) · Mn+1

Mixed labyrinth fractals generated by special cross patterns

Figure: Example: the first five elements of a sequence of special crosspatterns, where mk = 2k + 9, and ak = k + 4

Theorem
There exist sequences {Ak }∞k=1 of both horizontally and vertically
blocked labyrinth patterns, such that the limit set L∞ has theproperty that for any two points in L∞ the length of the arca ⊂ L∞ that connects them is finite. For almost all points x0 ∈ a(with respect to the length) there exists the tangent at x0 to thearc a.

Proposition
There exist sequences {Ak }∞k=1 of (both horizontally andvertically) blocked labyrinth patterns, such that the limit set L∞
has the property that for any two points in L∞ the length of thearc a ⊂ L∞ that connects them is infinite.

Theorem
There exist sequences {Ak }∞k=1 of both horizontally and vertically
blocked labyrinth patterns, such that the limit set L∞ has theproperty that for any two points in L∞ the length of the arca ⊂ L∞ that connects them is finite. For almost all points x0 ∈ a(with respect to the length) there exists the tangent at x0 to thearc a.

Proposition
There exist sequences {Ak }∞k=1 of (both horizontally andvertically) blocked labyrinth patterns, such that the limit set L∞
has the property that for any two points in L∞ the length of thearc a ⊂ L∞ that connects them is infinite.

Theorem
There exist sequences {Ak }∞k=1 of both horizontally and vertically
blocked labyrinth patterns, such that the limit set L∞ has theproperty that for any two points in L∞ the length of the arca ⊂ L∞ that connects them is finite. For almost all points x0 ∈ a(with respect to the length) there exists the tangent at x0 to thearc a.

Proposition
There exist sequences {Ak }∞k=1 of (both horizontally andvertically) blocked labyrinth patterns, such that the limit set L∞
has the property that for any two points in L∞ the length of thearc a ⊂ L∞ that connects them is infinite.

NEW! Very recent results:
On arclength in mixed labyrinth fractals
TheoremLet {Ak }k≥1 be a sequence of horizontally and vertically blocked
labyrinth patterns, such that the corresponding sequence of widths{mk }k≥1 satisfies the condition
X 1
k≥1

mk

= ∞.

Then, for all x, y ∈ L∞ with x 6= y the arc in L∞ that connects x
and y has infinite length.
Remark. The above condition is sufficient, but not necessary inorder to obtain a labyrinth fractal with all arcs having infinite
length.

NEW! Very recent results:
On arclength in mixed labyrinth fractals
TheoremLet {Ak }k≥1 be a sequence of horizontally and vertically blocked
labyrinth patterns, such that the corresponding sequence of widths{mk }k≥1 satisfies the condition
X 1
k≥1

mk

= ∞.

Then, for all x, y ∈ L∞ with x 6= y the arc in L∞ that connects x
and y has infinite length.
Remark. The above condition is sufficient, but not necessary inorder to obtain a labyrinth fractal with all arcs having infinite
length.

Very recent results: Supermixed labyrinth fractals

Figure: Three labyrinth patterns, from left to right: the unique pattern in{A1,1 } ∈ Ae1 , followed by the (two) patterns A2,1 , A2,2 ∈ Ae2 ,m1 = m2 = 4

Figure: The (width-homogeneous) supermixed labyrinth set W2 ,constructed with the help of the above patterns from Ae1 and Ae2 , that
can also be viewed as a 16-pattern

Very recent results: paths in supermixed labyrinth sets
Theorem
For all n ≥ 1,
M(n) = Qn,1 + · · · + Qn,sn+1 , sn+1 = #(Aen+1 )andM(n + 1) =

sn+1
X

n,h
Qn,h · Mn+1,h , Qn,h = (qi,j
)

h=1
n,h
where qi,jis the number of j-squares in the path of type i in
G(Wn ) which at the next step are “substituted” according to thepattern An+1,h .

Remark. For sn+1 = 1, (for some n ≥ 1), we have M(n) = Qn,1and thus we recover the formula M(n + 1) = M(n) · Mn+1 provenearlier for mixed labyrinth fractals.

Very recent results: paths in supermixed labyrinth sets
Theorem
For all n ≥ 1,
M(n) = Qn,1 + · · · + Qn,sn+1 , sn+1 = #(Aen+1 )andM(n + 1) =

sn+1
X

n,h
Qn,h · Mn+1,h , Qn,h = (qi,j
)

h=1
n,h
where qi,jis the number of j-squares in the path of type i in
G(Wn ) which at the next step are “substituted” according to thepattern An+1,h .

Remark. For sn+1 = 1, (for some n ≥ 1), we have M(n) = Qn,1and thus we recover the formula M(n + 1) = M(n) · Mn+1 provenearlier for mixed labyrinth fractals.

A. A. Potapov, W. Zhang, CIE International Conference on Radar(October 2016) : prototypes of ultra-wide band radar antennas based on
labyrinth fractals

A. A. Potapov, A. A. Potapov Jr., V. A. Potapov, Conference Paper(December 2017) : Fractal radioelements, devices and fractal systems forradar and telecommunications

Th a nkyo u !

What if the pattern is not blocked?

! If only one of the directions of the generating pattern isblocked, then there are pairs of points in the labyrinth fractals suchthat the length of the arc between them is finite.
Example:Lemma
The length of the arc in L∞ between any two distinct points
x, y ∈ L∞ is finite if and only if the straight line segment from x
to y is vertical and is contained in L∞ .

Wild labyrinth patterns/Wild labyrinth fractals

Figure: Examples: wild labyrinth patterns, both vertically andhorizontally blocked• tree ←→ connected graph• uniqueness of v/h exit pair ←→ existence of v/h exit pair
• corner property

Paths in wild labyrinth fractals. Example
For wild labyrinth fractals the Lemma about the path construction
does not hold in general: the squares in the shortest path from the
top exit to the bottom exit in G(W2 ) do not lie whithin the
shortest path from the top exit to the bottom exit in G(W1 )• in G(W1 ):1 = 15,left = 15,
1
1 =

1 =

right
= 13,
1
1 =

1 =9

-path in G(W2 ) is
7 1 + 2 1 + 1 + 1 + 1 + 1 = 7 · 13 + 2 · 15 + 4 · 9 = 157• the length of the “left” -path in G(W2 ) is
3 1 + 0 · 1 + 3 1 + 3 1 + 3 1 + 3 1 = 3 · 13 + 4 · 3 · 9 = 147• the length of the “right”