aarjaneiro/planetmath huggingface dataset:
linear function

linear function

Let

𝒮

1

=

(

𝒫

1

,

ℒ

1

)

and

𝒮

2

=

(

𝒫

2

,

ℒ

2

)

be two

near-linear spaces

.

Definition

. A

linear function

from

𝒮

1

to

𝒮

2

is a

mapping

on the points that sends lines of

𝒮

1

to lines of

𝒮

2

. In other words, a linear function is a

function

σ

:

𝒫

1

→

𝒫

2

such that

σ

⁢

(

ℓ

)

∈

ℒ

2

⁢

for every

⁢

ℓ

∈

ℒ

1

.

Here,

σ

⁢

(

ℓ

)

is the set

{

σ

⁢

(

P

)

∣

P

∈

ℓ

}

. A linear function is also called a

homomorphism

.

When both

𝒮

1

and

𝒮

2

are

linear spaces

, then

σ

being a linear function is equvalent to saying that

P

,

Q

are

collinear

iff

σ

⁢

(

P

)

,

σ

⁢

(

Q

)

are collinear.

If

𝒮

1

is a linear space, then so is

(

σ

⁢

(

𝒫

1

)

,

σ

⁢

(

ℒ

1

)

)

. This shows that if

σ

:

𝒮

1

→

𝒮

1

is onto,

𝒮

2

is a linear space if

𝒮

1

is.

Let

σ

:

𝒮

1

→

𝒮

2

be a

one-to-one

linear function. If points

P

1

≠

P

2

lie on line

ℓ

, then

σ

⁢

(

P

1

)

≠

σ

⁢

(

P

2

)

lie on

σ

⁢

(

ℓ

)

. This also shows that three

collinear points

in

𝒮

1

are mapped to three collinear points in

𝒮

2

. In

addition

, we have

|

ℓ

|

=

|

σ

⁢

(

ℓ

)

|

for any line

ℓ

in

𝒮

1

.

Definition

. When

σ

:

𝒮

1

→

𝒮

2

is a

bijection

whose

inverse

σ

-

1

is also linear, we say that

σ

is an

isomorphism

. When

𝒮

1

=

𝒮

2

=

𝒮

, we call

σ

an

automorphism

, or more commonly among geometers, a

collineation

, of the space

𝒮

.

Suppose

σ

:

𝒮

1

→

𝒮

2

is an isomorphism. For every point

P

, let

P

*

be the set of all lines

passing through

P

. Then

|

P

*

|

=

|

σ

⁢

(

P

)

*

|

for any point

P

in

𝒮

1

.

It is possible to have a

bijective

linear function whose inverse is not linear. For example, let

𝒮

1

be the space with two points

P

,

Q

with no lines, and

𝒮

2

the space with the same two points with line

{

P

,

Q

}

. Then the

identity function

on

{

P

,

Q

}

is a bijective linear function whose inverse is not linear. On the other hand, if the both spaces are linear, then the inverse is always linear.

Remark

. The usage of the term “linear function” differs from its more usual meaning as a

linear transformation

between

vector spaces

in the study of

linear algebra

.

References

1

L. M. Batten,

Combinatorics

of

Finite Geometries

, 2nd edition, Cambridge University Press (1997)

Title

linear function

Canonical name

LinearFunction

Date of creation

2013-03-22 19:14:46

Last modified on

2013-03-22 19:14:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51A45

Classification

msc 51A05

Classification

msc 05C65

Generated on Fri Feb 9 21:59:13 2018 by

LaTeXML
pencil

pencil

A

pencil

is a set of geometric objects, usually either

congruent

or

similar

to each other, that share a common incidence property. Below are some of the most commonly encountered pencils:

1.

A

pencil of lines

usually means a set of

straight lines

that are

incident

with one point. If the lines all lie in the same plane, the pencil is sometimes called a

flat pencil

.

2.

In some cases, a pencil of lines denotes a set of

parallel lines

in a plane. If a point of

infinity

is added to the plane, then we are back to the previous example.

3.

A pencil of circles can mean that either these circles all

intersect

at exactly one point (or share the same

tangent line

)

4.

A pencil of circles can also mean that the circles have two common points of

intersection

.

References

1

E. Artin,

Geometric Algebra

, Wiley-Interscience, Reprint (1988).

2

H. S. M. Coxeter,

Projective Geometry

, Springer-Verlag, 2nd Edition (2003).

Title

pencil

Canonical name

Pencil

Date of creation

2013-03-22 15:21:31

Last modified on

2013-03-22 15:21:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51A99

Related topic

FanOfLines

Related topic

HyperbolasOrthogonalToEllipses

Defines

flat pencil

Generated on Fri Feb 9 21:59:35 2018 by

LaTeXML
Pick’s theorem

Pick’s theorem

Let

P

⊂

ℝ

2

be a polygon with all vertices on

lattice points

on the grid

ℤ

2

. Let

I

be the number of lattice points that lie

inside

P

, and let

O

be the number of lattice points that lie on the boundary of

P

. Then the area of

P

is

A

⁢

(

P

)

=

I

+

1

2

⁢

O

-

1

.

In the above example, we have

I

=

5

and

O

=

13

, so the area is

A

=

10

⁤

1

2

; inspection shows this is true.

Title

Pick’s theorem

Canonical name

PicksTheorem

Date of creation

2013-03-22 12:46:58

Last modified on

2013-03-22 12:46:58

Owner

ariels (338)

Last modified by

ariels (338)

Numerical id

4

Author

ariels (338)

Entry type

Theorem

Classification

msc 51A99

Classification

msc 05B99

Classification

msc 68U05

Generated on Fri Feb 9 21:59:37 2018 by

LaTeXML
proof of Pick’s theorem

proof of Pick’s theorem

Pick’s

theorem

:

Let

P

⊂

ℝ

2

be a

polygon

with all vertices on lattice points on the grid

ℤ

2

. Let

I

be the number of lattice points that lie

inside

P

, and let

O

be the number of lattice points that lie on the

boundary

of

P

. Then the area of

P

is

A

⁢

(

P

)

=

I

+

1

2

⁢

O

-

1

.

To prove, we shall first show that Pick’s theorem has an

additive

character. Suppose our polygon has more than 3 vertices. Then we can divide the polygon

P

into 2 polygons

P

1

and

P

2

such that their

interiors

do not meet. Both have fewer vertices than

P

.

We claim that the

validity

of Pick’s theorem for

P

is

equivalent

to the validity of Pick’s theorem for

P

1

and

P

2

.

Denote the area, number of interior lattice points and number of boundary lattice points for

P

k

by

A

k

,

I

k

and

O

k

,

respectively, for

k

=

1

,

2

.

Clearly

A

=

A

1

+

A

2

.

Also, if we denote the number of lattice points on the edges common to

P

1

and

P

2

by

L

,

then

I

=

I

1

+

I

2

+

L

-

2

and

O

=

O

1

+

O

2

-

2

⁢

L

+

2

Hence

I

+

1

2

⁢

O

-

1

=

I

1

+

I

2

+

L

-

2

+

1

2

⁢

O

1

+

1

2

⁢

O

2

-

L

+

1

-

1

=

I

1

+

1

2

⁢

O

1

-

1

+

I

2

+

1

2

⁢

O

2

-

1

This proves the claim. Therefore we can triangulate

P

and it suffices to prove Pick’s theorem for

triangles

. Moreover by further

triangulations

we may assume that there are no lattice points on the boundary of the triangle other than the vertices.
To prove pick’s theorem for such triangles, embed them into

rectangles

.

Again by additivity, it suffices to prove Pick’s theorem for rectangles and rectangular triangles which have no lattice points on the

hypotenuse

and whose other two sides are

parallel

to the

coordinate

axes.
If these two sides have lengths

a

and

b

,

respectively, we have

A

=

1

2

⁢

a

⁢

b

and

O

=

a

+

b

+

1

.

Furthermore, by thinking of the triangle as half of a rectangle, we get

I

=

1

2

⁢

(

a

-

1

)

⁢

(

b

-

1

)

.

(Note that here it is essential that no lattice points are on the hypotenuse)
From these equations for

A

,

I

and

O

,

Pick’s theorem is satisfied for these triangles.

Finally for a rectangle, whose sides have lengths

a

and

b

,

we find that

A

=

a

⁢

b

I

=

(

a

-

1

)

⁢

(

b

-

1

)

and

O

=

2

⁢

a

+

2

⁢

b

.

From these Pick’s theorem follows for rectangles too.
This

completes

our proof.

Title

proof of Pick’s theorem

Canonical name

ProofOfPicksTheorem

Date of creation

2013-03-22 13:09:47

Last modified on

2013-03-22 13:09:47

Owner

giri (919)

Last modified by

giri (919)

Numerical id

5

Author

giri (919)

Entry type

Proof

Classification

msc 51A99

Classification

msc 05B99

Classification

msc 68U05

Generated on Fri Feb 9 21:59:39 2018 by

LaTeXML
homothetic

homothetic

Let

𝒫

and

𝒬

be

similar

polygons

,

(

P

1

,

P

2

,

…

,

P

n

)

and

(

Q

1

,

Q

2

,

…

,

Q

n

)

the respective vertices, so that corresponding vertices under

similarity

are listed in the same order.

We say that

𝒫

and

𝒬

are

homothetic

if all the lines

P

j

⁢

Q

j

are

concurrent

.

A

⁢

B

⁢

C

⁢

D

is homothetic to

A

′

⁢

B

′

⁢

C

′

⁢

D

′

.

The concurrence point of all lines is called the

homothety center

Notice that a necessary and

sufficient

condition for two similar polygons to be homothetic is that corresponding sides be

parallel

.

If

O

is the homothety center of two polygons

𝒜

and

ℬ

, then for any pair

P

,

Q

of corresponding points under the homothety (that is,

A

⁢

B

passes through

O

), the ratio

k

=

O

⁢

P

O

⁢

Q

is known as

homothety ratio

, and it coincides with the similarity ratio. It is customary to work with

directed segments

when talking of homothety, so the homothety ratio is a signed number.

When

k

is

positive

we speak of

direct homothety

and not only similarity is preserved, but pictures also have the same

orientation

(corresponding points to the upper part of one figure are on the upper part of the other).
However, if

k

is negative we have

reverse homothety

, and corresponding lines remain parallel but all orientations are reversed, so it looks like one picture is upside-down when compared to the other.

Direct homothety

Reverse homothethy

There is a special case where we have two

congruent

figures with the same orientation, in this case all the dashed lines are parallel, and we say that the homothety center is the

point at infinity

.

An alternate way of defining homothety is requiring the

existence

of some point

O

such that all the lines

P

⁢

O

where

P

is a point on a figure,

intersect

the other figure on a point

Q

in such way that all the ratios

O

⁢

P

/

O

⁢

Q

are the same.

Although both

definitions

are

equivalent

for polygons, the later has the benefit that it can be applied to arbitrary figures, and so we can talk about homothethy between arbitrary figures. For

instance

, it can be proved that any two circles are homotopic, in both direct and

inverse

ways, and the homothety centers can be located intersecting appropiated circle

tangents

when they exist.

Title

homothetic

Canonical

name

Homothetic

Date of creation

2013-03-22 15:04:06

Last modified on

2013-03-22 15:04:06

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 51Axx

Synonym

homothety

Related topic

SimilarityInGeometry

Related topic

SimilitudeOfParabolas

Defines

homothety center

Defines

homothety ratio

Defines

direct homothety

Defines

reverse homothety

Generated on Fri Feb 9 21:59:42 2018 by

LaTeXML
Bertrand’s problem

Bertrand’s problem

Given an

equilateral triangle

inscribed

on a circle, draw a chord on the circle at random.

Bertrand’s problem

, first posed by Joseph Bertrand in 1888, asks: what is the probability that that chord will be longer than a side of the

triangle

? Put another way, what is the probability that the length of a chord from a random single point along the

circumference

of the circle to a vertex of the triangle will form an angle from the intersection of the angle along the circle?

There are at least three different ways to randomly choose the chords: by choosing

midpoints

, by choosing

endpoints

, or by choosing chords. Thus the following solutions are obtained:

1. Randomly place a dot anywhere in the circle. Then draw the chord that has that dot as its midpoint. If the

distance

from the chord’s midpoint to the center of the circle is less than half the radius of the circle, then the chord will definitely be longer than a side of the triangle. The probability is then 25%.

2. Randomly place a dot anywhere on the circumference of the circle. Then draw a chord to connect that dot to any vertex of the triangle. It becomes visibly obvious that the midpoint of that chord has to fall somewhere between the two vertices other than the vertex we chose to connect our random dot to in order for the resulting chord to be longer than a side of the triangle. The probability is then approximately 33%.

3. Randomly draw a chord that is

parallel

to one of the sides of the triangle. If the chord falls entirely on what is not part of the triangle, then it is clear that it will be shorter than that side. The same is true for that reflected part on the other side of the triangle. The probability is then 50%.

Because three different results to this problem are obtained depending on the selection method, the problem is sometimes known as

Bertrand’s paradox

.

Title

Bertrand’s problem

Canonical name

BertrandsProblem

Date of creation

2013-03-22 16:24:18

Last modified on

2013-03-22 16:24:18

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

14

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51D20

Synonym

Bertrand’s paradox

Related topic

CircleLinePicking

Generated on Fri Feb 9 21:59:44 2018 by

LaTeXML
circle line picking

circle line picking

Circle line picking

is the activity of choosing two points on the

circumference

a unit circle and calculating probabilities and averages about the chord formed by connecting those two points.

The length of the

line segment

between the two points is

2

-

2

⁢

cos

⁡

θ

, where

θ

is the angle of the

central vertex

of a

triangle

formed by connecting the two points to the center. On average, this works out to

4

π

.

Title

circle line picking

Canonical

name

CircleLinePicking

Date of creation

2013-03-22 16:26:05

Last modified on

2013-03-22 16:26:05

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51D20

Related topic

BertrandsProblem

Generated on Fri Feb 9 21:59:46 2018 by

LaTeXML
circumscribe

circumscribe

To

circumscribe

a

regular polygon

(such as a

triangle

, square,

pentagon

, etc.) or a

cyclic quadrilateral

, is to enclose it in a circle so that the vertices of the

polygon

are on the

circumference

of the circle.

For example, a

circumscribed

square:

One more example, a circumscribed pentagon.

Title

circumscribe

Canonical name

Circumscribe

Date of creation

2013-03-22 16:24:32

Last modified on

2013-03-22 16:24:32

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

11

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51D20

Synonym

circumscribed

Synonym

circumscription

Related topic

RegularPolygonAndCircles

Related topic

Inscription

Generated on Fri Feb 9 21:59:48 2018 by

LaTeXML
happy ending problem

happy ending problem

The

happy ending problem

asks, for a given integer

n

>

2

, to find the smallest number

g

⁢

(

n

)

of points laid on a plane in

general position

such that a subset of

n

points can be made the vertices of a convex

n

-agon.

Figure 1:

Examples showing that

g

⁢

(

4

)

≤

5

.

It is

obvious

that for

n

=

3

, just three points in general position
are

sufficient

to create a

triangle

. For

n

=

4

, Paul Erdős and
Esther Klein (later Szekeres) determined that at least five points are
necessary, and Kalbfleisch later determined

g

⁢

(

5

)

=

9

. Much later,
George Szekeres (posthumously) and Lindsay Peters published a

computer

proof

[

6

]

that

g

⁢

(

6

)

=

17

.

For higher

n

, Erdős and George Szekeres in 1935 gave the upper bound

g

⁢

(

n

)

≤

(

2

⁢

n

-

4

n

-

2

)

+

1

.

Later, in 1961 they gave the lower bound

g

⁢

(

n

)

≥

1

+

2

n

-

2

.

New ideas for the upper bound were in the air in the late 1990s, with Chung and Graham showing that if

n

≥

4

, then

g

⁢

(

n

)

≤

(

2

⁢

n

-

4

n

-

2

)

,

while Kleitman and Pachter showed that then

g

⁢

(

n

)

≤

(

2

⁢

n

-

4

n

-

2

)

+

7

-

2

⁢

n

.

And Géza Tóth and Pavel Valtr in 1998 gave the upper bound

g

⁢

(

n

)

≤

(

2

⁢

n

-

5

n

-

3

)

+

2

,

which in 2005 they refined to

g

⁢

(

n

)

≤

(

2

⁢

n

-

5

n

-

3

)

+

1

.

References

1

F. R. K. Chung and R. L. Graham, Forced convex

n

-gons in the plane,

Discrete Comput. Geom.

19

(1996), 229–233.

2

P. Erdős and G. Szekeres, A combinatorial problem in

geometry

,

Compositio Math.

2

(1935), 463–470.

3

P. Erdős and G. Szekeres, On some extremum problems in

elementary

geometry,

Ann. Univ. Sci. Budapest Eötvös Sect. Math.

3–4

(1961), 53–62.

4

D. Kleitman and L. Pachter, Finding convex sets among points in the plane,

Discrete Comput. Geom.

19

(1998), 405–410.

5

W. Morris and V. Soltan, The Erdős-Szekeres problem on points in convex position – a survey,

Bull. Amer. Math. Soc.

37

(2000), 437–458.

6

L. Peters and G. Szekeres, Computer solution to the 17-point Erdős-Szekeres problem,

ANZIAM J.

48

(2006), 151–164.

7

G. Tóth and P. Valtr, Note on the Erdős-Szekeres

theorem

,

Discrete Comput. Geom.

19

(1998), 457–459.

8

G. Tóth and P. Valtr, The Erdős-Szekeres theorem: upper bounds and related results. Appearing in J. E. Goodman, J. Pach, and E. Welzl, eds.,

Combinatorial and computational geometry

, Mathematical Sciences Research Institute Publications

52

(2005) 557–568.

Title

happy ending problem

Canonical

name

HappyEndingProblem

Date of creation

2013-03-22 16:16:21

Last modified on

2013-03-22 16:16:21

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

13

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51D20

Synonym

happy end problem

Generated on Fri Feb 9 21:59:51 2018 by

LaTeXML
inscribe

inscribe

To

inscribe

a

polygon

(such as a

triangle

, square,

pentagon

, etc.) or circle is to enclose it in another polygon or circle so that the vertices of the polygon are on the

circumference

of the circle or sides of the outer polygon, or so that the circumference of the inner circle touches the sides of the enclosing polygon. In the specific case of a polygon inside of a circle, this is called to

circumscribe

.

Now, a couple of examples: First, a circle

inscribed

in a pentagon.

Next, another example of a pentagon, but inscribed with a triangle this time.

Title

inscribe

Canonical name

Inscribe

Date of creation

2013-03-22 16:24:35

Last modified on

2013-03-22 16:24:35

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51D20

Synonym

inscribed

Synonym

inscription

Related topic

RegularPolygonAndCircles

Related topic

Circumscribe

Generated on Fri Feb 9 21:59:53 2018 by

LaTeXML
lazy caterer’s sequence

lazy caterer’s sequence

Given a pancake (or a circle), how can one cut

n

pieces (not necessarily of the same size) with the minimum number of cuts? For example, to cut a pancake into four pieces, four cuts could be made, each starting at the center and going to the edge. But it would be much simpler to make just two cuts to cut it into four pieces.

The maximum number of pieces that can be created with a given number of cuts

n

is given by the formula

n

2

+

n

+

2

2

which gives the

lazy caterer’s

sequence

: 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, … (listed in A000124 of Sloane’s OEIS).

The numbers of this sequence are also called

central polygonal numbers

, and have applications in various other mathematical problems. Each of these numbers is 1 plus a

triangular number

.

Shel Kaphan, in a remark to the OEIS writes that ”when constructing a

zonohedron

, one zone at a time, out of (up to) 3-D non-intersecting

parallelepipeds

, the

n

th element of this sequence is the number of edges in the

n

th zone added with the

n

th layer of parallelepipeds.”

Title

lazy caterer’s sequence

Canonical name

LazyCaterersSequence

Date of creation

2013-03-22 16:16:54

Last modified on

2013-03-22 16:16:54

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51D20

Synonym

lazy caterers sequence

Synonym

circle cutting problem

Synonym

pancake cutting problem

Defines

central polygonal number

Generated on Fri Feb 9 21:59:55 2018 by

LaTeXML
length

length

The

length

of a

line segment

is the

distance

between its

endpoints

. Length may be measured in meters, yards, abstract units, etc. For example, in the following diagram

A

⁢

B

is one unit long,

B

⁢

C

is two units long,

C

⁢

D

is a fifth of a unit,

D

⁢

E

is four fifths of a unit,

A

⁢

C

is three units, etc.

The

concept

of length as defined above is a special case of a general concept called measure.

In two-dimensional space, length usually goes along the

x

axis while height goes along the

y

axis. The same holds in three-dimensional space.

For

triangles

,

pentagons

, and higher

n

-gons (

http://planetmath.org/BasicPolygon

), it is customary to refer to the length of any of its sides as the

length of the

. The length of a circle’s side is called its

circumference

.

In

set theory

,

length

refers to the number of

elements

a set (or one-dimensional array) has. This is also known as

cardinality

. For example,

{

2

,

5

,

11

,

23

,

47

}

has length

5

.

Title

length

Canonical

name

Length

Date of creation

2013-03-22 16:31:45

Last modified on

2013-03-22 16:31:45

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

17

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51D20

Classification

msc 51-00

Classification

msc 51M25

Related topic

Area2

Related topic

Volume2

Related topic

BasicPolygon

Defines

length of a

polygon

Defines

length of a set

Generated on Fri Feb 9 21:59:57 2018 by

LaTeXML
Motzkin number

Motzkin number

A

Motzkin number

for a given number

n

is the number of different ways of drawing non-intersecting chords on a circle between

n

points. The Motzkin numbers have very diverse applications in

geometry

,

combinatorics

and

number theory

. The first few Motzkin numbers are 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, (

sequence

A001006 in the OEIS).

The Motzkin number for

n

is also the number of

positive integer

sequences

n

-

1

long in which the opening and ending

elements

are either 1 or 2, and the

difference

between any two consecutive elements is -1, 0 or 1.

Also on the upper right quadrant of a grid, the Motzkin number for

n

gives the number of routes from coordinate (0, 0) to coordinate (

n

, 0) if one is allowed to move only to the right (either up, down or straight) at each step but forbidden from dipping below the

y

=

0

axis.

All together, there are at least fourteen different manifestations of Motzkin numbers in different branches of mathematics, as enumerated by Donaghey and Shapiro in their 1977 survey of Motzkin numbers.

Title

Motzkin number

Canonical

name

MotzkinNumber

Date of creation

2013-03-22 16:19:36

Last modified on

2013-03-22 16:19:36

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51D20

Generated on Fri Feb 9 21:59:59 2018 by

LaTeXML