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ALGORITHMICALLY FINDING THE IDENTITY OF THE BTW
SANDPILE GROUP
JACOB HAVEN AND ATTILA PÓR
Abstract. Algorithms to find the identity of the group of recurrent BTW
sandpiles are described. These algorithms are used to provide experimental
data, from which conjectures about the structure of the identity are tested.
1. Introduction
The BTW sandpile model was introduced by Bak, Tang and Wiesenfeld in
1988 [1] to study the 1/f noise. In their paper, they describe one, two and three
dimensional sandpiles as grids of heights (indexed by d-tuples, where d is the di-
mension), with each site toppling sand onto its 2d neighbors should it ever reach
a height of 2d. Since its introduction, the BTW sandpile model has been stud-
ied extensively by physicists for its demonstration of self-organized criticality, the
tendency to approach a critical state from many starting positions. Many mod-
ifications of the BTW model have been created, but the most important is the
generalization by Dhar in [5], [4], and [6] of the BTW model to the Abelian Sand-
pile Model (ASM). The ASM is defined for arbitrary directed graphs, represented
as toppling matrices, ∆, which describe the rules for when to topple at a site and
which sites (neighbors) to topple to. The algebra of the ASM can be studied to give
general mathematical results that may then be applied to special cases, such as the
BTW model. A good introduction to the general ASM, upon which the simplified
model here is based, is given in [8]. Most relevant to the topic is the finding that
the recurrent elements of the ASM have a group structure whose identity has been
studied, with some results, in [2] and [7].
Outline. The remainder of this article is organized as follows. Section 2 precisely
defines the specific BTW model studied and gives account of relevant previous
results. Section 3 describes the derivation of the algorithms used to find the identity
of the group of recurrent states. Section 4 analyses the identities produced to give
conjectures about their structure and fractal nature. Finally, Section 5 introduces
some further questions warranting future study.
2. BTW Sandpile Model on a Square Lattice
2.1. Sandpiles. We will be discussing the BTW sandpile model on the n × n grid,
so we shall define the set of indices of this grid. Let Nn be the set of all integers
from 1 to n. N2
n = Nn ×Nn is thus our set of indices. A sandpile on the n×n grid is
a height function from the indices of the grid to the natural numbers (starting with
0). Let Sn be the set of all n × n sandpiles. For an arbitrary sandpile, η ∈ Sn and
2000 Mathematics Subject Classification. 82B20.
1

2 JACOB HAVEN AND ATTILA P ÓR
a position x = (i, j) ∈ N2
n, η(x) = η(i, j) = z, where z ∈ N is the height (number
of sand grains) of the sandpile at x. A stable sandpile is one which has maximum
height 3. Let Ωn be the set of all stable sandpiles, which is clearly a subset of Sn.
Let Qn and Zn be sandpile sets defined similarly to Sn, with the sandpiles returning
rational numbers and integers respectively, instead of natural numbers. We assign
the names αnand ωnto the minimum and maximum stable sandpiles that return 0
and 3 respectively.
Nn = {k ∈ Z+
| k ≤ n} = {1, 2, . . . , n}
N2
n = Nn × Nn = {(1, 1), . . . , (n, n)}
Sn = {η | η : N2
n → N}
Ωn = {η | η : N2
n → {0, 1, 2, 3}}
Qn = {η | η : N2
n → Q}
Zn = {η | η : N2
n → Z}
∀x ∈ N2
n :
αn(x) = 0
ωn(x) = 3
(1)
We will now define sandpile equality and partial ordering
Definition 2.1 (Sandpile Comparison). For all η, ζ ∈ Qn:
η = ζ if and only if η(x) = ζ(x) for all x ∈ N2
n
The comparisons ≤, ≥, <, and > are defined similarly.
max(η) = η(xmax), where η(xmax) ≥ η(x), for all x ∈ N2
n
min(η) is defined similarly.
For convenience, we will define the standard matrix representations of an arbi-
trary sandpile to be the matrix of all values it returns.
Definition 2.2 (Matrix Form). For η ∈ Qn and {x1, x2, . . . , xn2 } = N2
n:
(2) mat(η) =



η(1, 1) . . . η(1, n)
...
...
...
η(n, 1) . . . η(n, n)



2.2. Toppling. For sandpiles on the grid, we are interested in the von Neumann
neighborhood, N(x), of each point x ∈ N2
n. This neighborhood contains all the
points directly above, below, and to the left or right of x. The number of neighbors
of x, | N(x)|, is thus four in the center of the grid, three on the edges, and two in
the corners. Also note that y ∈ N(x) if and only if x ∈ N(y).
Definition 2.3 (von Neumann Neighborhood). For all For all x = (x1, x2) ∈ N2
n
and y = (y1, y2) ∈ N2
n, let dp = p
|x1 − y1|p + |x2 − y2|p. Thus, d1 = |x1 − y1| +
|x2 − y2|
N(x) = {y ∈ N2
n | d 1xy = 1}
The model will be represented using the n2
×n2
lattice laplacian toppling matrix,
∆n
, which completely describes the relationships between each lattice point. A

FINDING THE SANDPILE IDENTITY 3
point relates to itself with the number of its neighbors should the boundary be
removed (i.e.: all 4s), and to its neighbors with -1. Notice that as neighbor relations
are symmetric (the graph is undirected), so is ∆n
Definition 2.4 (Lattice Laplacian Toppling Matrix). For all indices x ∈ N2
n, and
y ∈ N2
n:
∆n
x,y =



4 if x = y
−1 if y ∈ N(x)
0 otherwise
With this toppling matrix, we may describe a types of operation on sandpiles
called firing and toppling rules. A firing rule acts on single site, x, by removing four
grains of sand from it and adding one to each of its neighbors, or in other words,
it fires x. A toppling rule fires only unstable sites.
Definition 2.5 (Toppling Rules). A firing rule is an operation Fx : Zn → Zn
For all η ∈ Zn and x, y ∈ N2
n:
Fx(η)(y) = η(y) − ∆n
x,y
A toppling rule is an operation Tx : Sn → Sn such that for all η ∈ Sn:
Tx(η) =
Fx(η) if η(x) ≥ 4
η otherwise
By composing a finite number of toppling rules, we may obtain a toppling func-
tion that relaxes an arbitrary sandpile, η ∈ Sn to a unique stable sandpile, called
the relaxation of η.
Definition 2.6 (Toppling Function). The toppling function, T∆n : Sn → Ωn, is
defined for some minimal sequence (x1, x2, . . . , xN ), where xi ∈ N2
n, to be:
T∆n =
N
i=1
T xi = T xi ◦ T x2 ◦ . . . ◦ T xN
(x1, x2, . . . , xN ) is minimal in the sense that for η ∈ (Sn ⊕ Ωn) (i.e.: η is unstable).
T∆n (η) =
N−1
i=1
T xi (η) ∈ Ωn
For convenience, we will denote the relaxation of η ∈ Sn as η = T∆n (η).
Also, η = ζ may be denoted η → ζ.
It is proven in section 2.3 of [8] that T∆n is well defined by Definition 2.6, using
the fact the toppling rules are commutative (i.e.: Tx ◦ Ty = Ty ◦ Tx). It is also
worthwhile to note that Tx ◦T∆n = T∆n , for all x ∈ N2
n, as toppling rules only act
when a site is unstable. Following from this, we can see that T k
∆n = T∆n for all
k ≥ 1.

2.3. Addition. Let the operation of adding a single grain of sand at the site x ∈ N2
n
be denoted px : Qn → Qn. Let ax : Sn → Ωn, be the operation of adding a grain
of sand at x and allowing the sandpile to collapse. ax = T∆n ◦ px. Thus, for all
η ∈ Sn, px(η) → ax(η). Note that px and ax are both associative and commutative,
and that:
a
∆n
x,x
x =
y∈N(x)
a
−∆n
x,x
y
a4
x =
y∈N(x)
ay
(3)
The sum of two sandpiles is simply the sum of their heights at all grid points.
The sum with relaxation is the summation, followed by relaxation. Clearly, both
operations are associative and commutative.
Definition 2.7. Let us extend px, so that pk
x corresponds to adding k ∈ Q grains
of sand when k in nonnegative, and subtracting it is negative. For sandpiles η ∈ Sn
and ζ ∈ Sn, the summation of η and ζ is the result of summing the heights at all
points x ∈ N2
n.:
(η + ζ)(x) = η(x) + ζ(x) = ζ(x) + η(x)
η + ζ =


x∈N2
n
pζ(x)
x

 (η) ∈ Sn.
Scalar multiplication for sandpiles is defined similarly to scalar multiplication
for vectors. For k ∈ Q:
(k · η)(x) = k · η(x)
k · η =


x∈N2
n
pk·ζ(x)
x

 (η) ∈ Zn.
Subtraction of η ∈ Qn and ζinQn is defined as:
η − ζ = η + (−1) · ζ
Definition 2.8. For sandpiles η ∈ Sn and ζ ∈ Sn, the sum with relaxation is
η ⊕ ζ = η + ζ .
For convenience, let us also define the scalar multiplication by k ∈ Z+
k ⊗ η =
k η’s
η ⊕ . . . ⊕ η = k · η
Parentheses may be omitted because ⊕ is associative.
The following translations from operations to sandpile sums can prove useful:
∀η ∈ Qn px η = (px αn) + η(4)
∀η ∈ Sn ax η = (ax αn) ⊕ η(5)

Theorem 2.9. For all η ∈ Sn, and ζ ∈ Sn:
η ⊕ ζ =


x∈N2
n
aζ(x)
x

 (η)
Proof.
η ⊕ ζ = η + ζ =

T∆n
x∈N2
n
pζ(x)
x

 (η)
We can now use the property that T∆n = T k
∆n for all k ≥ 1 to obtain:
η ⊕ ζ =

T n2
∆n
x∈N2
n
pζ(x)
x

 (η) =


x∈N2
n
T∆n ◦ pζ(x)
x

 (η) =


x∈N2
n
aζ(x)
x

 (η).
Theorem 2.10. For η ∈ Sn and ζ ∈ Sn:
η ⊕ ζ = η ⊕ ζ = η ⊕ ζ
Proof. By Theorem 2.9:
η ⊕ ζ =


x∈N2
n
aζ(x)
x

 (η)
Notice that in this form, Equation 3 is the same as applying Tx to ζ, where ζ(x) ≥ 4.
Thus, we may apply any number of Tx’s to ζ and still obtain the same result. Thus:
η ⊕ ζ =


x∈N2
n
aT∆n (ζ)(x)
x

 (η) =


x∈N2
n
a ζ (x)
x

 (η) = η ⊕ ζ
We may now reverse the order order of η and ζ and apply the same reduction to
obtain:
η ⊕ ζ = η ⊕ ζ = η ⊕ ζ
Corollary 2.11. From Theorem 2.10, we can see that in a relaxed sum of sandpiles,
only the outer relaxation must remain, with relaxation of the addends being optional.
Thus, η + ζ + θ = η + ζ + θ = η + ζ + θ = η + ζ + θ =. . .
2.4. Firing Sandpiles. The set Qn is an n2
dimensional vectorspace over the
rational numbers. Similarly Zn ⊂ Qn is an n2
dimensional vectorspace over the
integers.
Let δn
x ∈ Zn be the ﬁring sandpile at x ∈ N2
n, corresponding to the xth row of
∆n
.
(6) ∀x, y ∈ N2
n δn
x (y) = ∆n
x,y
Let ϕ be the complete toppling operation on a sandpile that transforms it with
toppling matrix, ∆n
.

Definition 2.12. ϕ : Qn → Qn. For all η ∈ Qn:
ϕ(η) = (∆n
)T
η =
x∈N2
n
η(x) · δn
x =


x∈N2
n
Fx

 (η) ∈ Rn.
Since ∆n
is invertible, ϕ is a non-degenerate linear transformation of Qn and
∀η ∈ Qn ϕ-1
(η) = (∆n
)−1
η.
For any η ∈ Sn, ϕ-1
(η) gives the number of firings necessary at each point to
obtain the minimum sandpile.
(7)


x∈N2
n
Fϕ-1
(η)(x)
x

 (η) = αn
Lemma 2.13. η ∈ Q+
n if and only if ϕ-1
(η)(x) ∈ Q+
n .
Proof. ϕ-1
(η)(x) ∈ Q+
n implies η ∈ Q+
n as:
η =
x∈N2
n
ϕ-1
(η)(x) · δn
x ≥
x∈N2
n
δn
x ≥ 0
Now, we will prove ϕ-1
(η)(x) ∈ Q+
n . Assume to the contrary that m = min(ϕ-1
(η)) <
0. Let ma = {x ∈ N2
n | ϕ-1
(η)(x) = m}. If x ∈ ma then
η(x) = ϕ(ϕ-1
(η))(x) = 4 ϕ-1
(η)(x) −
y∈N(x)
ϕ-1
(η)(y)
= 4m −
y∈N(x)
ϕ-1
(η)(y) ≤ (4 − | N(x)|) · m ≤ 0
Thus, η(x) ≤ 0, with equality if and only if | N(x)| = 4 and ϕ-1
(η)(y) = m for all
y ∈ N(x). Since η(x) ≥ 0 we have N(x) ⊂ ma whenever x ∈ ma. Thus, ma = N2
n,
meaning for all y ∈ N2
n, η(y) = m and | N(y)| = 4. Since there exist points on the
edges with fewer than 4 neighbors, this is a contradiction. Thus, m ≥ 0.
Now that we have the operator ϕ-1
, how we can be represent sandpiles as combi-
nations of the rows of ∆n
is of interest. Thus, we define Dn ⊂ Zn to be all integer
combinations of the rows of ∆n
, i.e.: the rowspace of ∆n
over Z. Let D+
n = Dn ∩Sn.
Lemma 2.14. η ∈ Dn if and only if ϕ-1
(η) ∈ Zn.
Proof. η ∈ Dn means there exists some ζ ∈ Zn such that (∆n
)T
ζ = η. Thus,
ϕ(ζ) = η and η = ϕ-1
(ζ).
Following directly from Lemmas 2.13 and 2.14:
Corollary 2.15. ϕ-1
(η) ∈ Sn if and only if η ∈ D+
n
Lemma 2.16. If η ∈ Ωn and ζ ∈ D+
n then ϕ-1
(η) ≤ ϕ-1
(η ⊕ ζ).
Proof. For some firing sequence (x1, x2, . . . , xN ), Let ζ0 = ζ and ζi = ζi−1 − δn
xi
and let η0 = η + ζ0 and ηi = η + ζi for all 0 ≤ i ≤ N:
η0 = η + ζ, η1 = η + ζ − δn
x1
, . . . , ηN = η + ζ −
N
i=1
δn
xi
= η ⊕ ζ
ζi ∈ Dn for all 0 ≤ i ≤ N by the following induction: ϕ-1
(ζ0) = ϕ-1
(ζ) ∈ D+
n .
Assuming ϕ-1
(ζk) ∈ Dn, then ϕ-1
(ζk+1) = p−1
xk
ϕ-1
(ζk) ∈ Dn.

We show that ϕ-1
(ζi) ∈ Sn by induction.
From Corollary 2.15, ϕ-1
(ζ0) ∈ Sn.
Let us assume ϕ-1
(ζk) ∈ Sn for some 0 ≤ k < N.
As ϕ-1
(ζk+1) = p
(| N(xk+1)|−4)
xk+1 ϕ-1
(ζk), ϕ-1
(ζk+1) /∈ Sn if and only if ϕ-1
(ζk)(xk+1) <
4 − | N(xk+1)|) and ζk(xk+1) ≥ 4, and may thus be fired.
ζk(xk+1) =


y∈N2
n
ϕ-1
(ζk)(y)δn
y

 (xk+1)
= 4 · ϕ-1
(ζk)(xk+1) −


z∈N(xk+1)
ϕ-1
(ζk)(z)δn
z

 (xk+1)
≤ 4 · ϕ-1
(ζk)(xk+1) − ϕ-1
(ζk)| N(xk+1)|
≤ 4 · (4 − | N(xk+1|) − (4 − | N(xk+1|) · | N(xk+1)| − 1
= 16 − 8| N(xk+1)| + | N(xk+1)|2
− 1 ≤ 16 − (8) ∗ 2 + (2)2
− 1 = 3 < 4
Thus, if a firing at xk+1 would make ϕ-1
(ζk) /∈ Sn, that site is already stable,
and thus no firing will take place. Thus ϕ-1
ζk+1 ∈ Sn. And thus, by induction,
ϕ-1
ζN ∈ Sn.
As ϕ-1
(ζN ) ≥ αn,
ϕ-1
(η ⊕ ζ) = ϕ-1
(η + ζN ) ≥ ϕ-1
(η)
2.5. Group Properties. We will now define the reachability of a sandpile from
another sandpile. This coincides with the intuitive notion of a ”larger” sandpile,
up to relaxation.
Definition 2.17 (Reachability). A sandpile, η ∈ Ωn is reachable from ζ ∈ Sn if
and only if there there exists a sandpile θ ∈ Ωn (By Theorem 2.10, this is equivalent
to θ ∈ Sn), such that η = ζ ⊕ θ. This is denoted ζ → η.
ζ and η are said to communicate (ζ ∼ η) if and only if ζ → η and η → ζ.
Rechability may be used to define a class of stable sandpiles, known as reccurent
sandpiles, that are reachable from all stable sandpiles.
Definition 2.18 (Reccurent States). A recurrent sandpile, η ∈ Ωn, is one that is
reachable from all ζ ∈ Ωn. The set of all recurrent sandpiles is thus
Rn = {η ∈ Ωn | ∀ζ ∈ Ωn ζ → η}
Note that for all η ∈ Rn and ζ ∈ Rn, η ∼ ζ.
Two sandpiles are called equivalent if and only if there exists some sequence of
firings sandpile ϕ-1
(η − ζ) that transform between the η ∈ Qn and ζ ∈ Qn are
equivalent if and only if η − ζ ∈ Dn, denoted η ζ. Note that if ζ ∈ Ωn, η → ζ
implies η ζ, as some integer number of firings can performed on η to give ζ.
Corollary 2.19. If η ∈ Ωn, ζ ∈ Rn, and η ζ, then ϕ-1
(η) ≤ ϕ-1
(ζ). If η ∈ Rn,
then η = ζ.
Proof. For some θ ∈ D+
n , ζ = η ⊕ θ. Thus, ϕ-1
(ζ) = ϕ-1
(η ⊕ θ). By Lemma 2.16,
ϕ-1
(η) ≤ ϕ-1
(ζ)
If η ∈ Rn, then by the symmetry of , ϕ-1
(ζ) ≤ ϕ-1
(η). Thus, η = ζ.

Corollary 2.20. If η, ζ ∈ Ωn, η ζ, and ϕ-1
(η) ≤ ϕ-1
(ζ), then η /∈ Rn.
Proof. Assume to the contrary that η ∈ Rn. Thus, by Corollary 2.19, ϕ-1
(ζ)(x) ≤
ϕ-1
(η), which is a contradiction.
Theorem 2.21. Rnwith the operation ⊕ forms an abelian group.
Proof. Let η ∈ Rn and ζ ∈ Rn
(1) ⊕ must be associative.
By Corollary 2.11:
a ⊕ (b ⊕ c) = a + b + c = a + b + c = (a ⊕ b) ⊕ c
Thus ⊕ is associative
(2) ⊕ must be commutative.
η ⊕ ζ = η + ζ = ζ + η = ζ ⊕ η, thus ⊕ is commutative,
(3) Rn must be closed under addition:
By the definition of a recurrent sandpile: For any β1, β2 ∈ Ωn, there
exists θ1, θ2 ∈ Ωn such that:
η = β1 ⊕ θ1
ζ = β2 ⊕ θ2
Thus, η ⊕ ζ = (β1 ⊕ β2) ⊕ (θ1 ⊕ θ2), and (β1 ⊕ β2) → (η ⊕ ζ).
As β2 is varying over all values in Ωn, we may take it to be αn, and thus
get: (β1 ⊕ αn) → (η ⊕ ζ).
Thus, for all β1 ∈ Ωn, β1 → (η ⊕ ζ). Thus (η ⊕ ζ) ∈ Rn.
(4) There exists a unique −η ∈ Rn such that η ⊕ −η ⊕ ζ = η η ⊕ ζ = ζ As
proven by Creutz in [3], −η = (|∆n
|−1)⊗η, where |∆n
| is the determinant
of the toppling matrix and the number of recursive states.
(5) There exists an identity en ∈ Rn, with the property that en ⊕ ζ = ζ.
From the previous, we see that en = η η.
We can now extend our notation of scalar multiplication for recurrent states.
Definition 2.22 (Scalar Multiplication for Rn). For all η ∈ Rn and k ∈ Z+
:
0 ⊗ η = en
(−k) ⊗ η = k ⊗ −η
(8)
3. Algorithms to Find the Identity
Definition 3.1. jn ∈ Dn is the sandpile such that jn(x) = 4 − | N(x)| for all
x ∈ N2
n. This definition implies that ϕ-1
(jn)(x) = 1 for all x ∈ N2
n.
mat(jn) =







2 1 · · · 1 2
1 0 0 1
...
...
...
1 0 0 1
2 1 · · · 1 2







Lemma 3.2. For all η ∈ Rn, jn ⊕ η = η.

Proof. This follows from Corollary 2.19 as jn ⊕ η is a recurrent state and it is
equivalent to η.
Lemma 3.3. k ⊗ jn ∈ Rn for some k ∈ N.
Proof. Let ηi = i ⊗ jn for all i ∈ N. As ηi = ηi−1 ⊕ jn, the sequence ϕ-1
(ηi) is
monotonically increasing by Lemma 2.16. Since ηi ∈ Ωn and |Ωn| = 4n2
< ∞ the
sequence ϕ-1
(ηi) must eventually become constant
Theorem 3.4. k ⊗ jn = en for all k ≥ N. k, N ∈ N.
Proof. By the previous lemma for some N we have a = k ⊗ jn a recurrent state.
Since a ⊕ a = 2k ⊗ jn = a therefor a = en and the Theorem follows.
Theorem 3.4 leads directly to an algorithm (Algorithm 1) for finding the iden-
tity: simply start with jn and keep adding jn until no changes are made, which
indicates by Lemma 3.2 that a reccurent state has been reached. A similar algo-
rithm (Algorithm 2), starts with jn, and doubles (with toppling) until no changes
are made.
Let χn ∈ Sn be defined such that χn(x) = 4 for all x ∈ N2
n.
Theorem 3.5. en = 4 ⊗ (χn − χn )
Proof. Let a = χn − χn ∈ Tn. For any x ∈ N2
n the height a(x) = χn(x) −
χn (x) = 4 − χn (x) ≥ 4 − 3 = 1. Therefore 4 · a has height 4 at every position of
the grid and 4 · a is a recurrent state. Since it is also in Tn it is the unit en.
It is useful to analyse the identity as its toppling matrix, ϕ-1
(en). From ex-
perimental results, this has a convex paraboloid-like structure, especially in the
center, but it is much lower on the sides. Also of interest is the relationship
between ϕ-1
(en) and ϕ-1
(en+2). Let us define the maximum height on the side,
smax = max(ϕ-1
(en)(1, j)), j ∈ N. ϕ-1
(en+2)(i + 1, j + 1) ≈ ϕ-1
(en) + smax(i, j),
for 1 ≤ i, j ≤ n. Let us define τ1 to be this expansion along the sides of ϕ-1
(en).
τ1;n+2(i, j) =
ϕ-1
(en) + smax(i, j) if 2 ≤ i, j ≤ n
0 otherwise
(9)
en+2 = k · jn ⊕ ϕ(τ1)(10)
This provides a good estimate in the center of the sandpile, but toward the edges
it overcompensates and requires much toppling. Let us define a better estimate for
the sides of the sandpile, τ2, that expands along the center.
τ2;n+2(i, j) =



ϕ-1
(en)(i, j) if 1 ≤ i, j ≤ n/2 − 1
ϕ-1
(en)(i + 2, j + 2) if n/2 + 1n/2 ≤ i, j ≤ n ϕ-1
(en)(i ± 1, j)
if |i − ( n/2 + 1)| < 1
ϕ-1
(en)(i, j ± 1) if |j − ( n/2 + 1)| < 1
(11)
en+2 = k · jn ⊕ ϕ(τ2)
(12)

4. Results From Computational Model
As can be seen in Appendix B, the identities approach a stable, fractal state,
with much symmetry. Of particular note is how much 2’s and 3’s (green and
red) dominate. In general, there is a large square of 2’s at the center, with four
triangular patterns of 3’s radiating outward to the edges. e2n+1 is related to e2n
in the following way: after seperating e2n into it’s four main symmetrical regions
(top left, top right, bottom left, bottom right), create a single empty column and
a single empty row, both in the center. Place 0 in the very center, with a single
column and row cross of 1’s inside of the square of 2’s, and 2’s filling up the rest.
Let us define Cn(i) = {x ∈ Sn | en(x) = i ∈ {0, 1, 2, 3}}. From the previous,
|C2n+1(i)| ≥ C2n(i)| for all i ∈ {0, 1, 2}, and |C2n+1(3)| ≥ C2n(3)|. Graphs of all
|Cn| for 3 < n < 125 Let Cn = |Cn|
n2 be the normalized |Cn|, such that
3
i=0 Cn = 1.
5. Further Questions
Let us define a function σ : [0, 1]2
→ {0, 2, 3} (where [0, 1]2
is the unit square)
as follows:
Definition 5.1. For i = 2, 3, σ(x) := i if and only if there exists an ε > 0 and an
N ∈ N such that for all n ≥ N and all y ∈ N2
n with the property d2(x, y
n ) < ε,
en(y) = i.
σ(x) = i ⇐⇒ ∃ε > 0, N ∈ Z+
∀n ≥ N∀y ∈ N2
n d2(x,
y
n
) < ε → en(y) = i
σ(x) = 0 otherwise.
(13)
Using σ, we may now define subsets of [0, 1]2
.
Definition 5.2. Let Ai = {x | σ(x) = i}, for i = 1, 2, 3 and x ∈ [0, 1]2
. Both sets
A2 and A3 are open subsets of the unit square by definition. Let B2 ⊂ A0 and
B3 ⊂ A0 be the boundaries of A2 and A3, respectively.
Let B = B2 ∩ B3 ⊂ A0 be the common boundary of A2 and A3.
We conjecture that the Hausdorff dimension of B is greater then one, but smaller
then two.
References
[1] P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality, Physical review A 38 (1988),
no. 1, 364–374.
[2] S. Caracciolo, G. Paoletti, and A. Sportiello, Explicit characterization of the identity config-
uration in an Abelian sandpile model, Journal of Physics A Mathematical General 41 (2008),
5003.
[3] M. Creutz, Abelian Sandpiles, Computers in Physics 5 (1991), no. 2, 198.
[4] D. Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev. Lett. 64
(1990), no. 14, 1613–1616.
[5] D. Dhar and R. Ramaswamy, Exactly solved model of self-organized critical phenomena, Phys-
ical Review Letters 63 (1989), no. 16, 1659–1662.
[6] D. Dhar, P. Ruelle, S. Sen, and D.N. Verma, Algebraic aspects of abelian sandpile models,
Journal of Physics A: Mathematical and General 28 (1995), 805–831.
[7] Y. Le Borgne, On the identity of the sandpile group, Discrete Mathematics 256 (2002), no. 3,
775–790.
[8] R. Meester, F. Redig, and D. Znamenski, The abelian sandpile; a mathematical introduction,
Markov Processes and Related Fields 7 (2001), 509.

Appendix A. Algorithms
Algorithm 1 Find Identity by adding jn.
en ← jn
repeat
olden ← en
en ← en ⊕ jn
until olden = en
return en
Algorithm 2 Find Identity by starting with jn and doubling.
en ← jn
repeat
olden ← en
en ← 2 · en
until olden = en
return en
Appendix B. Identities with their toppling vectors and other Data
e150

e151
e152

[ϕ-1
(e150)]
ϕ-1
(e150)

ϕ-1
(e152) − τ1;152
ϕ-1
(e152) − τ2;152

|Cn|(i)for3 ≤ n ≤ 125 Even n are ﬁt with:
|Cn|(0) = 0.0638703x2
.10212
|Cn|(1) = 0.166812x1
.64857
|Cn|(2) = 0.642312x1
.86251
|Cn|(3) = 0.332719x2
.09833
Odd n are ﬁt with:
|Cn|(0) = 0.0565012x2.12427
|Cn|(1) = 0.439039x1.48653
|Cn|(2) = 0.674312x1.855
|Cn|(3) = 0.289106x2.12412

Cn(i) for 3 ≤ n ≤ 125 Even n are ﬁt with:
Cn(0) = 0.0403779x0.208401
Cn(1) = 0.197368x−0.372473
Cn(2) = 0.610848x−0.128
Cn(3) = 0.306736x0.117251
Odd n are ﬁt with:
Cn(0) = 0.0485335x0.158368
Cn(1) = 0.840877x−0.667482
Cn(2) = 0.563351x−0.104149
Cn(3) = 0.18834x0.22276
Gatton Academy of Mathematics and Science, Western Kentucky University
E-mail address: jacob.haven670@wku.edu
Department of Mathematics, Western Kentucky University
E-mail address: attila.por@wku.edu

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