The Electronic Geometry Textbook Project Xiaoyu Chen LMIB - Department of Mathematics Beihang University, China

Outline Motivation Motivation Objective - EGT Objective - EGT Problems Problems Classification Classification Organization Organization Summarization Summarization

Motivation An enormous amount of geometric results and knowledge has been accumulated An enormous amount of geometric results and knowledge has been accumulated It is possible for computer to solve lots of geometric problems automatically It is possible for computer to solve lots of geometric problems automatically NO software system has been developed for managing geometric knowledge efficiently NO software system has been developed for managing geometric knowledge efficiently

Objective - EGT The Electronic Geometry Textbook (EGT) will integrate geometric knowledge and software modules for computer-supported geometric problem solving, to realize efficient management of knowledge in the form of textbook. The Electronic Geometry Textbook (EGT) will integrate geometric knowledge and software modules for computer-supported geometric problem solving, to realize efficient management of knowledge in the form of textbook. Textbook is the main representation of systematical knowledge Textbook is the main representation of systematical knowledge EGT is virtually a dynamic software system EGT is virtually a dynamic software system

Objective - continued EGT’s main feature is that EGT’s main feature is that all the geometric knowledge in it can be processed by available software modules for geometric computing and reasoning geometric computing and reasoning interactive and automated dynamic diagram generation interactive and automated dynamic diagram generation visualization of geometric objects etc. visualization of geometric objects etc. and the results will also display in the textbook

Objective - continued The texts can be translated from one language to another automatically. The texts can be translated from one language to another automatically. We can edit the textbook, such as searching knowledge, automatically checking the consistency while modifying the texts, re-organizing the structure, re-typesetting the textbook etc. We can edit the textbook, such as searching knowledge, automatically checking the consistency while modifying the texts, re-organizing the structure, re-typesetting the textbook etc. Even new knowledge can be added into the textbook automatically. Even new knowledge can be added into the textbook automatically.

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

Knowledge management The management of the whole textbook data can be divided into five steps: The management of the whole textbook data can be divided into five steps: Classification: The textbook can be divided into many different parts. The classification is helpful for organization. Classification: The textbook can be divided into many different parts. The classification is helpful for organization. Organization: Decide the logical structure of the textbook, in another word, to decide each part’s position in the textbook Organization: Decide the logical structure of the textbook, in another word, to decide each part’s position in the textbook Representation: Implement the classification and logical structure, represent the data on computer Representation: Implement the classification and logical structure, represent the data on computer

Knowledge management Manipulation: Manipulation: Operate the textbook, such as modifying the texts, re-organizing the structure, searching knowledge and so on Computation / Deduction: Computation / Deduction: Process the knowledge, such as computing, proving, drawing graphs with other software systems automatically etc.

Classification The first step: classify the texts according to their purposes “segment”: describes a set of texts with meaning “class” : describes a set of segments with common purpose Every segment in each class can be considered as an object Textbook can be divided into index class and content class Textbook can be divided into index class and content class Index class: decides the whole logical structure of the textbook, both global and local Index class: decides the whole logical structure of the textbook, both global and local Content class: contains all the texts in the textbook Content class: contains all the texts in the textbook

Classification Content class contains specification class and geometric knowledge class Content class contains specification class and geometric knowledge class Specification class: Specification class: background: to introduce the background of certain knowledge background: to introduce the background of certain knowledge explanation: just to explain the meaning of certain knowledge explanation: just to explain the meaning of certain knowledge example: to demonstrate certain knowledge with examples example: to demonstrate certain knowledge with examples graph: to demonstrate certain knowledge in the form of visualization graph: to demonstrate certain knowledge in the form of visualization proof/computing: to demonstrate the theorems, lemmas, formula deductions etc. proof/computing: to demonstrate the theorems, lemmas, formula deductions etc.

Classification of geometric knowledge is the main task. Classification of geometric knowledge is the main task. Geometric knowledge class: contains all the geometric knowledge segments Geometric knowledge class: contains all the geometric knowledge segments definition class definition class Lemma class Lemma class theorem class theorem class axiom class axiom class formula class etc. formula class etc. But this is not enough to organize them But this is not enough to organize them

Classification - knowledge For more detailed classification, we have to specify the structures of the geometric statements A statement of geometric knowledge is constructed by connecting many concepts with connectives like “implying”, “and”, “equivalence” etc. A statement of geometric knowledge is constructed by connecting many concepts with connectives like “implying”, “and”, “equivalence” etc. 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 circle Geometric quantities: length of a segment, area of a circle Quantity relation (function): the proportion of two segments etc. Quantity relation (function): the proportion of two segments etc.

Classification - knowledge As we all know, a definition is to state a new concept using defined concepts. The classification of concept class is equal to the classification of definition class. As we all know, a definition is to state a new concept using defined concepts. The classification of concept class is equal to the classification 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 objects like point, line, triangle and circle. We can also classify the knowledge by geometric objects like point, line, triangle and circle. For example, considering the hiberarchy of these geometric objects, the classification will be: For example, considering the hiberarchy of these geometric objects, the classification will be: The first class containing the knowledge related line, such as definition of point and line, the relations between point and line, line and line, and related theorems etc. The first class containing the knowledge related line, such as definition of point and line, the relations between point and line, line and line, and related theorems etc.

Classification - knowledge The second class containing the definition of triangle and the relations between point and triangle, triangle and triangle and other related knowledge. The third class containing the knowledge related circle. How to specify a “classification rule” for classifying the geometric knowledge automatically ? According to the classification, we can organize these segment objects to construct the textbook According to the classification, we can organize these segment objects to construct the textbook

Organization The second step: decide the “positions” of all the segments in the textbook. Main problem is deciding the “order” of the segments, especially the “order” of knowledge segments. Main problem is deciding the “order” of the segments, especially the “order” of knowledge segments. Because the other segments related with a knowledge segment are ordered around the knowledge in a normal way

Order of knowledge segments This “order” is “linear”: one by one from the beginning of the textbook to the end. This “order” is “linear”: one by one from the 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 knowledge segment must be given before this segment. 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 automatically. it is possible to check some ordering errors automatically.

Order of knowledge segments So, we need find more “rules” to decide the order of knowledge segments, just as a human textbook-writer does. So, we need find more “rules” to decide the order of knowledge segments, just as a human textbook-writer does. For example, we can consider our index (chapter-section) as a rule for the “order”. For example, we can consider our index (chapter-section) as a rule for the “order”. What kinds of rules are needed, and how can we find them? This is what I am thinking about. What kinds of rules are needed, and how can we find them? This is what I am thinking about.

Summarization The EGT is a dynamic software system The EGT is a dynamic software system The EGT contains systematical geometric knowledge for learning The EGT contains systematical geometric knowledge for learning