1. Introduction
As is well known, the theory of retractions is always one of interesting topics in Euclidian and Non-Euclidian space and it has been investigated from the various viewpoints by many branches of topology and differential geometry El-Ahmady [1-3].
Most folding problems are attractive from a pure mathematical standpoint, for the beauty of the problems themselves. The folding problems have close connections to important industrial applications Linkage folding has applications in robotics and hydraulic tube bending. Paper folding has application in sheet-metal bending, packaging, and air-bag folding El-Ahmady [2,3]. Following the great Soviet geometer El-Ahmady [4], also used folding to solve difficult problems related to shell structures in civil engineering and aero space design, namely buckling instability El-Ahmady [4].
Isometric folding between two Riemannian manifold may be characterized as maps that send piecewise geodesic segments to a piecewise geodesic segments of the same length El-Ahmady [2-4]. For a topological folding the maps do not preserves lengths El-Ahmady [5,6], i.e. A map
, where M and N are 
-Riemannian manifolds of dimension m, n respectively is said to be an isometric folding of M into N, iff for any piecewise geodesic path
, the induced path 
is a piecewise geodesic and of the same length as
, If 
does not preserve length, then 
is a topological folding El-Ahmady [7,8].
A subset 
of a topological space 
is called a retract of 
if there exists a continuous map 
such that 
where A is closed and 
is open El-Ahmady [9-11]. Also, let 
be a space and 
a subspace. A map 
such that 
for all 
is called a retraction of 
onto 
and 
is called a retract of
. This can be restated as follows. If 
is the inclusion map, then 
is a map such that 
If, in addition, 
we call r a deformation retract and 
a deformation retract of 
Another simple-but extremely useful-idea is that of a retract. If A, 
then 
is a retract of 
if there is a commutative diagram.
If 
and 
then f is a retract of g if there is a commutative diagram El-Ahmady [3,7], Arkowitz [12], Naber [13,14], Reid [15], Shick [16] and Strom [17].
2. Main Result
2.1. On a Closed Interval
In what follows, we discuss the retractions, let the closed interval be 
since the closed interval I is closed then firstly take a point 
of it to make a retraction, 
is open. Consider some types of retractions of
.
If 
then we can get
.
Now, we are in a position to formulate the following two theorems.
Theorem 1. All types of retraction of a closed interval are semi-open set or open set or zero-space.
Theorem 2. The limit of retraction of closed interval is a zero-manifold.
Now, we are going to discuss the deformation retract of the closed interval. Let 
be the open interval, and then the homotopy map is defined as
where 
then we present the following cases of deformation retracts
where 
and
From the above discussion, we obtain the following theorem.
Theorem 3. The deformation retracts of a closed interval gives semi-open set, open set and zero-dimensional space.
Now, we are going to discuss the folding 
of closed interval I.
Let
, where 
An isometric folding of closed interval I into itself may by
The deformation retracts of the folded closed interval 
into the folded retraction 
is
with
,
where 
and 
The deformation retract of the folded closed interval 
into the folded retraction 
is
The deformation retract of the folded closed interval 
into the folded retraction 
is
Then, the following theorem has been proved.
Theorem 4. The deformation retract of the folded closed interval into the folded retractions is the same as or different from the deformation retract of the closed interval into the retractions
Proposition 1. If the retraction of the closed interval 
is 
and the folding of (
) into itself is
, then there are commutative diagram between retraction and folding such that
Proof. Let a retraction
, be a retraction of 
into
. Also, let the folding 
is 
and the folding 
be
. Then we have the retraction 
such that
Proposition 2. The relation between the retraction of the closed interval and the limit of folding discussed from the following commutative diagram.
Proof. Let a retraction
, be a retraction of 
into
. Also, let the limits of folding are given by
and
.
Then we have the retraction of the zero dimensional manifold is the identity map, i.e. 
such that
2.2. On a Cartesian Product of Closed Interval
In this position, we introduce the retraction of Cartesian product of closed interval. Consider two closed intervals
. The Cartesian product is defined as
The retraction is defined as. Consider the square 
with vertices 
removing only one vertex 
then the retraction is given by,
.
Also, removing two adjacent vertices is equivalent to removing an edge of the square, and then the retraction is defined as follows,
Moreover, removing two non-adjacent vertices gives a retraction, which is directly the zero-dimensional manifold,
In what follows, we discuss the deformation retract of the square 
as follows. The deformation retract of the square 
is defined by
where 
is the closed interval
. Then we have the following cases of deformation retract. The deformation retract of the square 
onto a 
is given by
where 
and 
The deformation retract of the square 
onto closed interval 
will be
The deformation retract of the square 
onto zero-points is
From all the above discussion, we arrive to the following theorem.
Theorem 5. The limit of retractions sequence of the square is the 0-dimensional manifold. Also, the deformation retract of the square is either subsquare or zero-dimensional manifold.
Proposition 3. If the retraction of the square 
is
and the folding of 
into itself is
Then there are commutative diagrams between retractions and foldings such that
Also
And also
Proposition 4. The relation between the retraction of the square 
and the limit of folding discussed from the following commutative diagrams.
Also
Again
where the limit of the folding of the Cartesian product of 
is not equal to the Cartesian product of their limits.
Proposition 5. If the deformation retract of 
where 
and the retraction of 
is 
,
and the folding of 
into itself is 
Then there are induces deformation retractions, and folding such that the following diagram is commutative.
Proof. Let the deformation retract of 
is
the folding of
, and
are defined by 
also
, the deformation retract of 
is
and the retractions of
, and 
are given by
,
. Hence, the following diagram is commutative
Also, the end of limits of the folding and the end of limits of retractions of 
induces the 0-dimensional space which is a point and in this case the retraction and folding of 0-dimensional space coincide.
Proposition 6. The limit of the folding of 0-dimensional space M is 0-dimensional space.
Proof. Let 
be an n-dimensional space, consider the limit of the folding
,
, then
but if M has 0-dimension. Then
. Since 
, then 
.
Theorem 6. 
is a strong deformation retract of
.
Proof. Let
, where
, 
is a strong deformation retracting of
, 
into
. To be specific, the k-homotopy D is assumed:
, 
, and
, and
.
Let
, be defined as
Also
.
Then, 
is a strong deformation retract of
.
Proposition 7. The retraction of 
is a two-dimensional manifold and the limit of foldings is a one-manifold.
Proof. If A is a retraction of
, then either dimension A = dimension 
or dimension A ≠ dimension
, but in this case dimension 
not invariant. Then 
is the same dimension of I2. But, the limit of the foldings of 2-dimensional manifold into itself is a manifold of dimension n − 1. Then, the limit of foldings is a one-manifold.
2.3. On a Circle
Theorem 7. If 
has the fixed point property, then 
is not a retract of
.
Proof. Let 
has the fixed point property. Observe that 
certainly does not have the fixed point property since, for example, the antipodal map-
is continuous, but has no fixed points. Then 
can therefore not be a retract of
.
Proposition 8. If 
has the fixed point property, then 
is not a retract of
, where
.
Proof. Since
, and 
does not have the fixed point property. Then 
does not have the fixed point property,
. Then 
is not a retract of
.
Theorem 8. If 
is a 0-manifold, then 
is a retract of
, such that
,
, and
where 
and
.
Proof. Now, let
, 
,
be the retraction map of 
defined as
. Let the inclusion map of
, where
, is
, the retractions of 
and 
are defined by 
and
, where
,
is the inclusion map of
, the retraction of 
is
, also the retractions of 
and 
are given by 
and
.
Hence the following diagram is commutative:
Proposition 9. The relation between the retraction, the limit of the folding and the inclusion map of circle 
discussed from the following commutative diagram
where
and
The purpose of this position is to introduce the relation between the deformation retract and folding of the circle, the parametric equation of the open circle in the plane is given by
.
Now consider some types of retractions of the circle
, if
then we can get
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
Now, we are going to discuss the deformation retract of the circle. Let 
be the open circle, then the homotopy map is defined as
where
,
Then we have the following cases of deformation retracts.
where 
and
,
,
,
Now, we are going to discuss the folding 
of the circle
. let
, where
.
An isometric folding of the circle 
into itself may by defined by
The deformation retracts of the folded circle 
into the folded retraction 
is
with
where
and 
Then, the following theorem has been proved.
Theorem 9. Under the defined folding and any folding homeomorphic to this type of folding, the deformation retracts of the folded circle into the folded retractions 
is the same as the deformation retracts of the circle into the retractions.
If the folding be defined by
.
The isometric folded of the circle 
is defined as
The deformation retract of the folded circle 
into the folded retraction 
is
with
The deformation retract of the folded circle 
into the folded retraction 
is
Then, the following theorem has been proved.
Theorem 10. Under the defined folding and any folding homeomorphic to this type of folding, the deformation retract of the folded circle into the folded retractions is different from the deformation retract of the circle into the retractions.
Proposition 10. If the retraction of the circle 
is:
, and the folding of
into itself is
).
Then there are commutative diagrams between retraction and folding such that:
Proof. Let a retraction
, be a retraction of 
into
. Also, let the folding 
is given by
and
, also,
such that
Let 
where
is the circle with center 
and radius
. The intersection of all circles is denoted by 0. Let
, be the retraction map of 
such that
,
,
,
,
,
,
,
,
Hence, we can introduce the following theorems:
Theorem 11. Any circle 
with center 
and radius
, where
, is a retract of
.
Proof. Let 
and 
is a retract of
, where
. Also, If 
, is the inclusion map, the retractions of 
and 
are defined by
and
, the inclusion map of 
is
, and the retractions of
, and 
are given by
, 
and
. Hence, the following diagram is commutative:
Proposition 11. Any circle 
with center 
and radius
, where
, is a retract of
.
Theorem 12. Any circle 
with center 
and radius
, where
, is a retract of
.
Proof. Since 
is a retract of 
and any circle 
in 
with center 
and radius
, where
, is a retract of
, then any circle 
with center 
and radius
, where
, is a retract of
. Also, Since
, 
is a retract of 
and any circle 
with center 
and radius
, where
, is a retract of
, then any circle 
with center 
and radius
, were
, is a retract of
.
Theorem 13. Any retract of circle
, 
in 
with center 
and radius
, where
, is a retract of
.
Proof. Let 
is a retract of
, then there is a continuous map
, 
, where
. Then the circle 
in 
with center 
and radius
, where
, is a retract of
, then there is a continuous map
,
, where
.Then, 
is a continuous map. Also,
. Then any retract of circle 
in with center 
and radius
where
, is a retract of
.
3. Conclusion
In this paper we achieved the approval of the important of the curves in the Euclidean space by using some geometrical transformations. The relations between folding, retractions, deformation retracts, limits of folding and limits of retractions of the curves in the Euclidean space are discussed. New types of minimial retractions on curves in the Euclidean space are deduced.