The The EElectronic lectronic GGeometry eometry TTextbook Projectextbook Project

Xiaoyu ChenXiaoyu Chen

LMIB - Department of MathematicsLMIB - Department of MathematicsBeihang University, ChinaBeihang University, China

OutlineOutline

MotivationMotivation Objective - EGTObjective - EGT ProblemsProblems

Classification Classification OrganizationOrganization

SummarizationSummarization

MotivationMotivation

An enormous amount of geometric results and An enormous amount of geometric results and knowledge has been accumulated knowledge has been accumulated

It is possible for computer to solve lots of It is possible for computer to solve lots of geometric problems automaticallygeometric problems automatically

NO software system has been developed for NO software system has been developed for managingmanaging geometric knowledge efficiently geometric knowledge efficiently

Objective - EGTObjective - EGT The Electronic Geometry Textbook (EGT) The Electronic Geometry Textbook (EGT)

will will integrateintegrate geometric knowledge and geometric knowledge and software modules for computer-supported software modules for computer-supported geometric problem solving, to realize efficient geometric problem solving, to realize efficient management of knowledge in the form of management of knowledge in the form of textbook.textbook.

Textbook is the mainTextbook is the main representation ofrepresentation of systematical knowledgesystematical knowledge

EGT is virtually a dynamic software systemEGT is virtually a dynamic software system

Objective - continuedObjective - continued

EGT’s main EGT’s main featurefeature is that is that

all the geometric knowledge in it can be all the geometric knowledge in it can be processed by available software modules for processed by available software modules for geometric computing and reasoning geometric computing and reasoning interactive and automated dynamic diagram interactive and automated dynamic diagram

generation generation visualization of geometric objects etc.visualization of geometric objects etc.

and the results will also display in the textbookand the results will also display in the textbook

Objective - continuedObjective - continued

The texts can be translated from one language to The texts can be translated from one language to another automatically.another automatically.

We can edit the textbook, such as searching We can edit the textbook, such as searching knowledge, automatically checking the consistency knowledge, automatically checking the consistency while modifying the texts, re-organizing the while modifying the texts, re-organizing the structure, re-typesetting the textbook etc.structure, re-typesetting the textbook etc.

Even new knowledge can be added into the textbook Even new knowledge can be added into the textbook automatically.automatically.

PossibilityPossibility

Because the statements of mathematical Because the statements of mathematical textbooks, not only the knowledge, are normal textbooks, not only the knowledge, are normal and there are logical rules to constrain and there are logical rules to constrain textbook’s structure. textbook’s structure.

Knowledge managementKnowledge management

The management of the whole textbook data The management of the whole textbook data can be divided into five steps:can be divided into five steps:

ClassificationClassification: The textbook can be divided into : The textbook can be divided into many different parts. The classification is helpful many different parts. The classification is helpful for organization. for organization.

OrganizationOrganization: D: Decide the logical structure of the ecide the logical structure of the textbook, in another word, to decide each part’s textbook, in another word, to decide each part’s positionposition in the textbook in the textbook

RepresentationRepresentation: I: Implement the classification and mplement the classification and logical structure, represent the data on computerlogical structure, represent the data on computer

Knowledge managementKnowledge management

ManipulationManipulation: :

Operate the textbook, such as modifying the Operate the textbook, such as modifying the texts, re-organizing the structure, searching texts, re-organizing the structure, searching knowledge and so onknowledge and so on

Computation / DeductionComputation / Deduction::

PProcess the knowledge, such as computing, rocess the knowledge, such as computing, proving, drawing graphs with other software proving, drawing graphs with other software systems automatically etc.systems automatically etc.

ClassificationClassification

The first step: The first step: classifyclassify the texts according to their the texts according to their purposes purposes

““segment”: describes a set of texts with meaningsegment”: describes a set of texts with meaning

““class” : describes a set of segments with common purpose class” : describes a set of segments with common purpose

Every segment in each class can be considered as an objectEvery segment in each class can be considered as an object Textbook can be divided into Textbook can be divided into indexindex classclass and and contentcontent

classclass Index classIndex class: decides the whole logical structure of the : decides the whole logical structure of the

textbook, both global and localtextbook, both global and local Content classContent class: contains all the texts in the textbook: contains all the texts in the textbook

ClassificationClassification Content class contains Content class contains specification classspecification class and and

geometric knowledge classgeometric knowledge class Specification class:Specification class:

backgroundbackground: to introduce the background of certain : to introduce the background of certain knowledgeknowledge

explanationexplanation: just to explain the meaning of certain : just to explain the meaning of certain knowledgeknowledge

exampleexample: to demonstrate certain knowledge with : to demonstrate certain knowledge with examplesexamples

graphgraph: to demonstrate certain knowledge in the form of : to demonstrate certain knowledge in the form of visualizationvisualization

proof/computingproof/computing: to demonstrate the theorems, lemmas, : to demonstrate the theorems, lemmas, formula deductions etc.formula deductions etc.

Classification of geometric knowledge is the Classification of geometric knowledge is the main task.main task.Geometric knowledge class: contains all the Geometric knowledge class: contains all the

geometric knowledge segmentsgeometric knowledge segments

definitiondefinition class classLemmaLemma class classtheoremtheorem class classaxiomaxiom class classformulaformula class etc. class etc.

But this is not enough to organize them But this is not enough to organize them

Classification - knowledgeClassification - knowledge

For more detailed classification, we have to For more detailed classification, we have to specify specify the the structures of the geometric structures of the geometric statementsstatements

A statement of geometric knowledge is constructed A statement of geometric knowledge is constructed by connecting many by connecting many conceptsconcepts with connectives like with connectives like “implying”, “and”, “equivalence” etc.“implying”, “and”, “equivalence” etc.

These concepts have four kinds:These concepts have four kinds: Geometric objects: point, line, triangle, circle etc.Geometric objects: point, line, triangle, circle etc. Object relation: parallel, intersect etc. Object relation: parallel, intersect etc. Geometric quantities: length of a segment, area of a circleGeometric quantities: length of a segment, area of a circle Quantity relation (function): the proportion of two Quantity relation (function): the proportion of two

segments etc.segments etc.

Classification - knowledgeClassification - knowledge As we all know, a definition is to state a new concept As we all know, a definition is to state a new concept

using defined concepts. The using defined concepts. The classification of concept classification of concept classclass is equal to the is equal to the classification of definition classclassification of definition class..

Geometric objects are the basic of other concepts.Geometric objects are the basic of other concepts. We can also classify the knowledge by geometric objWe can also classify the knowledge by geometric obj

ects like point, line, triangle and circle.ects like point, line, triangle and circle.

For example, considering the hiberarchy of these geoFor example, considering the hiberarchy of these geometric objects, the classification will be:metric objects, the classification will be:

The first class containing the knowledge related line, sThe first class containing the knowledge related line, such as definition of point and line, the relations betweuch as definition of point and line, the relations between point and line, line and line, and related theorems een point and line, line and line, and related theorems etc.tc.

Classification - knowledgeClassification - knowledge

The second class containing the definition of triangle anThe second class containing the definition of triangle and the relations between point and triangle, triangle and trid the relations between point and triangle, triangle and triangle and other related knowledge.angle and other related knowledge.

The third class containing the knowledge related circle.The third class containing the knowledge related circle.

How to specify a “classification rule” for classifying the How to specify a “classification rule” for classifying the geometric knowledge automatically ? geometric knowledge automatically ?

According to the classification, we can organize these seAccording to the classification, we can organize these segment objects to construct the textbookgment objects to construct the textbook

OrganizationOrganization

The second step: decide the “positions” of all the The second step: decide the “positions” of all the segments in the textbook.segments in the textbook.

Main problem is deciding the “order” of the Main problem is deciding the “order” of the segments, segments, especially the “order” ofespecially the “order” of knowledge knowledge segmentssegments..

Because the other segments related with a Because the other segments related with a knowledge segment are ordered around the knowledge segment are ordered around the knowledge in a knowledge in a normalnormal way way

Order of knowledge segmentsOrder of knowledge segments This “order” is “linear”: one by one from the This “order” is “linear”: one by one from the

beginning of the textbook to the end.beginning of the textbook to the end. A rule for the “order” of knowledge segments:A rule for the “order” of knowledge segments:The definition segment for every concept in one The definition segment for every concept in one knowledge segment must be given before this knowledge segment must be given before this segment.segment.By this rule,By this rule,

computer can only decide a part of the “order” .computer can only decide a part of the “order” . it is possible to check some ordering errors it is possible to check some ordering errors

automatically.automatically.

Order of knowledge segmentsOrder of knowledge segments

So, we need find more “rules” to decide the So, we need find more “rules” to decide the order of knowledge segments, just as a human order of knowledge segments, just as a human textbook-writer does.textbook-writer does.

For example, we can consider our index For example, we can consider our index (chapter-section) as a rule for the “order”.(chapter-section) as a rule for the “order”.

What kinds of rules are needed, and how can we What kinds of rules are needed, and how can we find them? This is what I am thinking about.find them? This is what I am thinking about.

SummarizationSummarization

The EGT is a dynamic software systemThe EGT is a dynamic software system

The EGT contains systematical geometric The EGT contains systematical geometric knowledge for learningknowledge for learning