Smarandache geometries

Smarandache geometries

An axiom is said

smarandachely denied

(S-denied) if in the same space the axiom behaves differently (i.e., validated and invalided; or only invalidated but in at least two distinct ways).

A

Smarandache geometry

(SG) is a

geometry

which has at least one smarandachely denied axiom (1969).

Thus, as a particular case,

Euclidean

, Lobachevsky-Bolyai-Gauss, and Riemannian geometries may be united altogether, in the same space, by some SGs. These last geometries can be partially Euclidean and partially non-Euclidean.

The novelty of the SG is the fact that they introduce for the first time

the degree of

negation

in geometry

, similarly to the degree of falsehood in fuzzy or

neutrosophic logic

. For example an axiom can be denied in percentage of 30

Also SG are defined on multispaces, i.e. unions of Euclidean and non-Euclidean

subspaces

, or unions of distinct non-Euclidean spaces.

As an example of S-denying, a

proposition

ϕ

, which is the

conjunction

of a set

ϕ

i

of propositions, can be

invalidated in many ways

if it is minimally

unsatisfiable

, that is, such that the conjunction of any

proper subset

of the

ϕ

i

is satisfied in a

structure

, but

ϕ

itself is not.

Many axioms have one of the following forms in the sequential logic:

Let

n

,

m

⩾

1

be two integers, also

A

=

A

1

×

A

2

×

⋯

×

A

n

and

B

=

B

1

×

B

2

×

⋯

×

B

m

.

Let

x

=

(

x

1

,

x

2

,

⋯

,

x

n

)

and

y

=

(

y

1

,

y

2

,

⋯

,

y

m

)

, also

𝑃

⁢

(

x

)

and

𝑄

⁢

(

x

)

be relationships on

x

and

𝑅

⁢

(

x

,

y

)

a relationship between

x

and

y

. Then:

a

1

∀

x

∈

A

,

∃

and is unique

y

∈

B

, such that

𝑅

⁢

(

x

,

y

)

.

a

2

If

x

∈

A

such that

𝑃

⁢

(

x

)

, then

𝑄

⁢

(

x

)

.

a

3

∀

x

∈

A

one has

𝑃

⁢

(

x

)

.

One can invalidate in many ways the first class of axioms

a

1

as follows:

1.1.

∃

x

∈

A

and

∃

finitely many (at least two)

y

∈

B

such that

𝑅

⁢

(

x

,

y

)

.

1.2.

∃

x

∈

A

and

∃

infinitely many

y

∈

B

such that

𝑅

⁢

(

x

,

y

)

.

1.3.

∃

x

∈

A

such that

∄

⁢

y

∈

B

for which

𝑅

⁢

(

x

,

y

)

.

One can invalidate in many ways the second class of axioms

a

2

as follows:

2.1.

∃

a unique

x

∈

A

such that

𝑃

⁢

(

x

)

but

¬

⁢

𝑄

⁢

(

x

)

.

2.2.

∃

finitely many (more than one)

x

∈

A

such that

𝑃

⁢

(

x

)

but

¬

⁢

𝑄

⁢

(

x

)

.

2.3.

∃

infinitely many

x

∈

A

such that

𝑃

⁢

(

x

)

but

¬

⁢

𝑄

⁢

(

x

)

.

2.4.

∀

x

∈

A

such that

𝑃

⁢

(

x

)

one has

¬

⁢

𝑄

⁢

(

x

)

.

One can invalidate in many ways the third class of axioms

a

3

as follows:

3.1.

∃

x

∈

A

and is unique such that

¬

⁢

𝑃

⁢

(

x

)

.

3.2.

∃

finitely many

x

∈

A

such that

¬

⁢

𝑃

⁢

(

x

)

.

3.3.

∃

infinitely many

x

∈

A

such that

¬

⁢

𝑃

⁢

(

x

)

.

Not all axioms can be smarandachely denied.

Here it is an example of what it means for

an axiom to be invalidated in

multiple

ways

[

2

]

:
As a particular axiom let’s take Euclid’s Fifth Postulate. In Euclidean or parabolic geometry a line has one

parallel

only through a given point. In Lobacevskian or

hyperbolic geometry

a line has at least two parallels through a given point. In Riemannian or elliptic geometry a line has no parallel through a given point. Whereas in Smarandache geometries there are lines which have no parallels through a given point and other lines which have one or more parallels through a given point (the fifth postulate is invalidated in many ways).

Therefore, the Euclid’s Fifth Postulate (which asserts that there is only one parallel

passing through

an

exterior

point to a given line) can be invalidated in many ways, i.e.

Smarandachely denied

, as follows:

- first invalidation: there is no parallel passing through an exterior point to a given line;

- second invalidation: there is a finite number of parallels passing through an exterior point to a given line;

- third invalidation: there are infinitely many parallels passing through an exterior point to a given line.

References

1

S. Chimienti, M. Bencze,

Smarandache Paradoxist Geometry

, Bulletin of Pure and Applied Sciences, Vol. 17E, No. 1, 123-1124, 1998.

2

H. Iseri,

Smarandache Manifolds

, Am. Res. Press, 2002.

http://www.gallup.unm.edu/ smarandache/Iseri-book.pdf

The book is also online..

3

L. Kuciuk, M. Antholy

An

Introduction

to Smarandache Geometries

, JP Journal of Geometry and Topology, Vol. 5, No. 1, 77-82, 2005.

Presented at

New Zealand Mathematics Colloquium

, Massey University, Palmerston North, New Zealand, December 3-6, 2001.

4

Linfan Mao,

An introduction to Smarandache geometries on maps

, 2005 International Conference on

Graph Theory

and

Combinatorics

, Zhejiang Normal University, Jinhua, Zhejiang, P. R. China, June 25-30, 2005.

5

Linfan Mao,

Automorphism Groups

of Maps, Surfaces and Smarandache Geometries

, partially post-doctoral research for the Chinese Academy of Science, Beijing, 2005.

http://www.gallup.unm.edu/ smarandache/Linfan.pdf

Second book which is online..

6

F. Smarandache,

Paradoxist Mathematics

, in

Collected Papers (Vol. II), Kishinev University Press, Kishinev, 5-28, 1997.

http://www.gallup.unm.edu/ smarandache/CP2.pdf

Third book which is online..

Title

Smarandache geometries

Canonical name

SmarandacheGeometries

Date of creation

2013-03-22 15:28:24

Last modified on

2013-03-22 15:28:24

Owner

jonnathan (5141)

Last modified by

jonnathan (5141)

Numerical id

7

Author

jonnathan (5141)

Entry type

Definition

Classification

msc 53A35

Related topic

FlorentinSmarandache

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