Geometry Presentation

в_Простран.ppt

Images :      Вращения_picks.mws     Сечения.mws     Цилидр_поверхн_РИС.mws      Цилиндр_кругов_РИС.mws      Цилиндр_наклонн_РИС.mws

Programs :      Вращения_прогр.mws      Цилидр_поверхн_прогр.mws      Цилиндр_кругов+СЕч_прогр.mws      Цилиндр_наклонн_прогр.mws

hi.please help me for solve this equation

i encounter with error''

Error, (in StringTools:-IsPrefix) second argument must be a string''

equations which be solved attached as pdf file

thanks

Kernel4.mw

root.pdf

Download Kernel4.mw

Hi

First question is, given a list of some positive integers, how can I normalize this list? Normalize here is in the sense that, for example, if 

L := [1,3,4,4,5,7,7,7,8]

then there really are only 6 different integers appear, so I would like to assign each part an integer from 1 to 6 in ascending order. So 1 becomes 1, 3 becomes 2, 4 becomes 3, 5 becomes 4, 7 becomes 5 and so on. Normalized list will be

NormalizedL := [1,2,3,3,4,5,5,5,6]

Second question is given a list, let's say [1,3,1,3,2,2,4,4], how can I normalize it in a similar way but now we assign each integer upon occurrence of a part. So [1,3,1,3,2,2,4,4] will be [1,2,1,2,3,3,4,4]. This is necessary for me because lists have repeating parts. 

Another example will be [2,4,4,1,2,2,3,3] will be [1,2,2,3,1,1,4,4]. 

Thank you 

My query is related to Maple packages "Lie algebra" and "DifferentialGeometry", I wonder can these packages help me to check isomophism between to Lie algebras, for example, suppose I have a 4-dim Lie algebra with commutation relations:

[e2, e3] = e3, [e2, e4] = - e4, [e3, e4] = - e1 

and another 4-dim Lie algebra with commutation relations

[e2, e3] = e1, [e4, e2] = e2, [e3, e4] = - e3

Can I check for isomophism between these Lie algebras if there is any using Lie algebra package ?

Regards

Hello! 

In an assigment I have been asked to use the Eigenvectors command to find the eigenvalues and eigenvectors of a particular matrix. 

As highlighted in the following image, my questions are:

1. What is the meaning of the suffix "+0.I"? Does it mean that there are further decimal digits which are not displayed?

2. How do the first and third eigenvalues, which are equal, result in different eigenvectors? As per my understanding, equal eigenvalues should have equal corresponding eigenvectors. Please help.

I need to plot a path with curvilinear coordinates. Knowing the curvature theta and length s of  parts that compose it, I found the equations of x and y as a function of the parameter s that goes from zero to the total length.
I make an example below:

for 0 <s <2,     x = s     and

                     y = 0 (straight line)

for 2 <s <2 + Pi,        x = 2 + 2 * sin ((s-2) / 2)      and

                              y = 2 * (1-cos ((s-2) / 2)

Now I want to find the corresponding plot of x (s), y (s) (without showing s as coordinate) but I do not know what commands I can use. Can you suggest me something? Thank you

I'm tryinv to find the contents of specific bins in a maple histogram. I don't find this elementary 'function' anywhere in the help.

How do I do it? Surely it's possible w/o taking a ruler to the screen...

Dasayeva Diana, 6th form

кит_Аним_Диана.mws

When I used exportplot in Maple2015 , the GIF file made easily after Enter. But when I use Engine.evaluate in Java jOpenMaple. it works for every command but exportplot. It didn't create any GIF file. I'm working on project must be use this for create animation by Maple but now that make big trouble. Any one to help this ?

VeHinhNemXien := proc(Alpha,Vbd)
  local Y,V0,alpha,X,ball,Xmax,bgr;
  with(plottools):
  with(plots):
  Y := unapply(V0*sin(alpha)*X/(V0*cos(alpha)) - 1/2*9.8*(X/(V0*cos(alpha)))^2,alpha,V0,X);
  ball := proc(x,y) plots[pointplot]([[x,y]],color=red,symbol=solidcircle,symbolsize=40) end proc;
  Xmax := 2*Vbd^2*sin(Alpha)*cos(Alpha)/9.8;
  bgr := plot(Y(Alpha,Vbd,X),X=0..Xmax,linestyle=[2]);
  animate(ball,[X,Y(Alpha,Vbd,X)],X=0..Xmax,scaling=constrained,labels=["Độ xa","Độ cao"],frames=60,background=bgr);
  exportplot(FileTools:-JoinPath([FileTools:-TemporaryDirectory(), "dothi.gif"]), animate(ball,[X,Y(Alpha,Vbd,X)],X=0..Xmax,scaling=constrained,labels=["Độ xa","Độ cao"],frames=60,background=bgr), gif);
end proc:

save VeHinhNemXien , "D:\\VeHinhNemXien.m";

in java file

String a[];
        a = new String[1];
        a[0] = "java";
        t = new Engine(a, new EngineCallBacksDefault(), null, null);

        t.restart();
        t.evaluate("read \"resources/VeHinhNemXien.m\";");

        t.evaluate(....query to call VeHinhNemXien to draw plot).

I checked carefully. When call it on Maple, it created GIF, but not in java. I checked queryString carefully.

New Info. I find that when I t.evaluate(....query to call VeHinhNemXien to draw plot) . this code make java stop there. that mean no code after this line can run.

Russian children work with Maple - view

http://geodromchik.blogspot.ru

//sites.google.com/site/geodromchic

POSSIBILITIES OF USING OF COMPUTER IN MATHEMATICS

AND OTHER APPLICATIONS IN INCLUSIVE EDUCATION

Alsu Gibadullina, math teacher

Secondary and high school # 57, Kazan, Russia

In recent years Russia actively promoted and implemented the so-called inclusive education (IE). According to the materials of Alliance of human rights organizations “Save the children”: "Inclusive or included education is the term used to describe the process of teaching children with special needs in General (mass) schools. In the base of inclusive education is the ideology that ensures equal treatment for everybody, but creates special conditions for children with special educational needs. Experience shows that any of the rigid educational system some part of the children is eliminated because the system is not ready to meet the individual needs of these children in education. This ratio is 15 % of the total number of children in schools and so retired children become separated and excluded from the overall system. You need to understand that children do not fail but the system excludes children. Inclusive approaches can support such children in learning and achieving success. Inclusive education seeks to develop a methodology that recognizes that all children have different learning needs tries to develop a more flexible approach to teaching. If teaching and learning will become more effective as a result of the changes that introduces Inclusive Education, all children will win (not only children with special needs)."

There are many examples of schools that have developed their strategy implementation of IE, published many theoretical and practical benefits of inclusion today. All of them have common, immaterial character. There is no description of specific techniques implementing the principles of IO in the teaching of certain disciplines, particularly mathematics. In this paper we propose a methodology that can be successfully used as in “mathematical education for everyone", also for the development of scientific creativity of children at all age levels of the school in any discipline.

According to the author, one of the most effective methodological tools for education is a computer mathematics (SCM, SSM). Despite the fact that the SCM were created for solving problems of higher mathematics, their ability can successfully implement them in the school system. This opinion is confirmed by more than 10–year-old author's experience of using the package Maple in teaching mathematics. At first it was just learning the system and primitive using its. Then author’s interactive demonstrations, e-books, programs of analytical testing were created by the tools packages. The experience of using the system Maple in teaching inevitably led to the necessity to teach children to work with it. At first worked a club who has studied  the principles of the package’s work, which eventually turned into a research laboratory for the use of computer technology. Later on its basis there was created the scientific student society (SSS) “GEODROMhic" which operates to this day. The main idea and the ultimate goal of SSS – individual research activities on their interests with the creation of the author electronic scientific journals through the use of computer mathematics Maple. The field of application of the package was very diverse: from mathematics to psychology and cultural phenomena. SSS’s activity is very high: they are constantly and successfully participate in intellectual high-level activities (up to international). Obviously, not every SSS’s member reaches high end result. However, even basic experience in scientific analysis, modeling, intelligent  using of the computer teaches the critical thinking skills, evokes interest to new knowledge, allows you to experience their practical value, gives rise to the development of creative abilities. As a result, the research activity improves intellectual culture, self-esteem and confidence, resistance to external negative influence. It should be noted, however, that members of the scientific societies are not largely the so-called "gifted", than ordinary teenagers with different level of intellectual development and mathematical training. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models, which contributes to the development of mathematical thinking.

From 2007 to 2012. our school (№. 57 of Kazan) was the experimental platform of the Republican study SKM (Maple) and other application software in the system of school mathematical education under the scientific management of Professor Yu. G. Ignatyev of Kazan state University (KF(P)U).

Practical adaptation of computer mathematics and other useful information technologies to the educational process of secondary schools passed and continues to work in the following areas:

1. The creation of a demonstration support of different types of the lessons;
2. The embedding of computing to the structure of practical trainings;
3. In the form of additional courses - studying of computer applications through which you can conduct a research of the mathematical model and create animated presentation videos, web-pages, auto-run menu;
4. Students’ working on individual creative projects:

• construction of computer mathematical models;
• creating author's programs with elements of scientific researches;
• students create interactive computer-based tutorials;
• creation of an electronic library of creative projects;

1. The participation of students  in the annual competitions and scientific conferences for students;
2. The accumulation and dissemination of new methodological experience.

Traditionally, the training system has the structure: explanation of a new →  the solution of tasks→ check, self-test and control → planning of the new unit  with using analysis. However, the main task types: 1) elementary, 2) basic, 3) combined, 4) integrated, 5) custom. With the increasing the level of training a number of basic tasks are growing and some integrated tasks become a class of basic. Thus, the library for basic operations is generated. The decision of the educational task occurs on the way of mastering the theoretical knowledge of mathematical modeling: 1) analysis of conditions (and construction drawing), 2) the search for methods of solution, 3) computation, 4) the researching.

To introduce computer mathematics in this training system, you can:

• At demonstrations. For example, with Maplе facilities you can create a step-by-step interactive and animated images, which are essentially the exact graphic interpretation of mathematical models.
• If we have centralized collective control.
• If students have individual self-control.
• In the analysis of the conditions of the problem, for the construction and visualization of its model, the study of this model.
• In the computations.
• In practical training of different forms.
• In individual projects with elements of research.

In the learning process with the use of computer mathematics in the school a library of themed demonstrations, tasks of different levels and purpose, programs, analytical testing, research projects is generated. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models. Addiction to them processed in the course of working with them it’s unobtrusive, naturally, organically.

Mathematical modelling (MM) is increasingly becoming an important component of scientific research. Today's powerful engineering tools allow to carry out numerous computer experiments, deep and full enough of exploring the object, without significant cost painless. Thus provided the advantages of theoretical approach, and experiment. The integration of information technology and      MM method is effective, safe and economical. This explains its wide distribution and makes unavoidable component of scientific and technical progress.

Modeling is a natural process for people, it is present in any activity. The introduction with nature by man occurs through constant  modeling of situation, comparing with the basic models and past experience by them. Method for modeling, abstraction as a method of understanding the world is therefore  an effective method of learning. Training activities associated with the creative transformation of the subject. The main feature of educational training activities is the systematic solution of the educational problems. The connection of the principles of developmental education, mathematical modeling, neurophysiological mechanisms of the brain and experience with Maple leads to the following conclusions: the method of mathematical modeling is not only scientific research but also the way of development of thinking in general; computer and mathematical environment (Maple), which is a powerful tool for scientific simulation can be considered as the elementary analogue of the brain. These qualities of computer mathematics led to the idea of using it not only as an effective methodological tool but as a means of nurturing the thinking and development of mental functions of the brain. To study this effect the school psychologist conducted a test, which confirm the observations: the dynamics of intellectual options students  who working with Maple compares favorably with peers. In the process of doing computer math, in particular Maple, are involved in complex different mental functions. It is in the inclusion of all mental functions is the essence of integration of learning, its educational character. And this, in turn, contributes to the solution of moral problems.

Long-term work with computer mathematics led to the idea to use it as a tool for psychological testing. One of the projects focuses on the psychology and contains authors Maple–tests to identify the degree of development of different mental functions. Interactive mathematical environment  gives a wide variability and creative testing capabilities. Moreover, Maple–test can be used not only as diagnostic but also as educational, and corrective. This technology was tested in one of psycho neurological dispensaries a few years ago.

Currently, one of the author's students, the so-called "homeworkers", the second year is a young man with a diagnosis, categories F20, who does not speak and does not write independently. It was         impossible to get feedback from him and have basic training until then author have begun to apply computer-based tools, including system Maple. Working with the computer tests and mathematical objects helps to see not only the mental and even the simplest thinking movement, but also emotional movement!

In general, the effectiveness of the implementation in the structure of educational process of secondary school of new organizational forms of the use of computers, based on the application of the symbolic mathematics package Maple, computer modeling and information technology, has many aspects, here are some of them:

• goals of education and math in particular;
• additional education;
• methodical and professional opportunities;
• theoretical education;
• modeling;
• scientific creativity;
• logical language;
• spatial thinking, the development of the imagination;
• programming skills;
• the specificity of technical translation;
• differentiation and individualization of educational process;
• prospective teaching, the continuity of higher and secondary mathematics education;
• development of creative abilities, research skills;
• analytical thinking;
• mathematical thinking;
• mental diagnosis;
• mental correction.

       According to the author, the unique experience of the Kazan 57–th school suggests that computer mathematics (Maple) is the most effective also universal tool of new methods of inclusion. In recent decades, there are more children with a specific behavior, with a specific perception, not able to focus, with a poor memory, poor thinking processes. There are children, emotionally and intellectually healthy, or even ahead of their peers in one team together with them. High school should provide all the common core learning standards. It needed a variety of programs and techniques, as well as specialists who use them. Due to its remarkable features, computer mathematics, in particular Maple, can be used or be the basis of the variation of methods of physico-mathematical disciplines of inclusive education.

Hi, I want to ask. the maple program that i have done have something wrong somewhere.

for an example changes basis

 from e_{1}e_{1}=e_{1} , e_{1}e_{2}=e_{2} maple program reading :A1 : (1,1,1)=1,B1: (1,2,2)=1

to e_{2}e_{2}=e_{2} , e_{2}e_{1}=e_{1} maple program reading :A2: (2,2,2)=1,B2: (2,1,1)=1

these changes basis above are 2 operation, left product and right product

A1 and A2 are left product, while B1 and B2 are right product,

i need to make A1 isomorphic to A2, and B1 isomorphic to B2.

by using maple program, i should get identity in matrix form 2x2

[0 1] but i get [0         1]

[1 0],            [C_{21} 0],

For isomorphism, the determinant should not be zero

here's are my maple program:

>isom := proc (A1, A2, B1, B2, n)

local i, j, k, s, r, eqns, t, TEST, BChange, sols, m, S1, S2, C;

C := matrix(n, n);

BChange := matrix(n, n);

TEST := 0; eqns := {};

for i to n do for j to n do for m to n do

S1 := sum(A1[i, j, k]*C[k, m], k = 1 .. n); S2 := sum(C[i, r]*(sum(A2[r, s, m]*C[j, s], s = 1 .. n)), r = 1 .. n);

eqns := `union`(eqns, {S1 = S2})

end do end do end do;

for i to n do for j to n do for m to n do

S1 := sum(B1[i, j, k]*C[k, m], k = 1 .. n); S2 := sum(C[i, r]*(sum(C[j, s]*B2[r, s, m], s = 1 .. n)), r = 1 .. n);

eqns := `union`(eqns, {S1 = S2})

end do end do end do;

sols := [solve(eqns)];

t := nops(sols);

for i to t do for j to n do for k to n do

BChange[k, j] := subs(sols[i], C[k, j])

end do end do;

if simplify(linalg:-det(BChange)) <> 0 then print("BChange", BChange);

print("s1", S1); print("s2", S2); print("The det is", simplify(linalg:-det(BChange)));

TEST := 1 end if end do;

if TEST = 0 then print("These two algebras are not isomorphic")

end if end proc

input maple program:

> DENDA1 := array(sparse, 1 .. 2, 1 .. 2, 1 .. 2, [(1, 1, 1) = 1]);
> DENDB1 := array(sparse, 1 .. 2, 1 .. 2, 1 .. 2, [(1, 2, 2) = 1]);
> DENDA2 := array(sparse, 1 .. 2, 1 .. 2, 1 .. 2, [(2, 2, 2) = 1]);
> DENDB2 := array(sparse, 1 .. 2, 1 .. 2, 1 .. 2, [(2, 1, 1) = 1]);
> isom(DENDA1, DENDA2, DENDB1, DENDB2, 2);

Hi All, 

I'm using the Physics package, which enables GR calculations, ie defining metrics and tensor algebra. 

Was just curious if it were possible to add a perturbation to the metric when calculating Ricci and Christoffels. 

I would like something like 

g_[] = g1_[mu,nu] + h[mu,nu] 

And then do a calculation like, 

Ricci[]. 

I know this would be possible if I define everything and re-write the calculations for calculating Ricci, i.e

Define(g1[mu,nu], h[mu,nu]); 

and the proceed with GR calculations to find Ricci, however was hoping there was an easier way to do this. 

Any help is appreciated. 

Thanks guys. 

Good evening all,

How can I plot a straightline with points 

LogAt = - 0.097,  -0.20, -0.22, -0.25, -0.30 ,-0.40, -0.45, -1.01 and

t = 0, 20, 40, 60, 80, 100, 120, 140

Where LogA[t] on y-axis and t on x-axis. 

I have try this before . 

plot([[0,-0.097], [20,-0.20], [40,-0.22], [60,-0.25], [80,-0.30], [100,-0.40],  [120,-0.45],  [140,-1.01]]);

Now in English   KozlovaAV.PDF

In Russian

Авторский опыт использования математической системы Maple и других компьютерных инструментов в школьном научном обществе

 Арина Козлова

E-mail: k_arina99@mail.ru; МБОУ «Школа  № 57» Кировского района г.Казани, 10 класс

 Научный руководитель –

Гибадуллина Алсу, учитель математики МБОУ «Школа  № 57» Кировского района г.Казани;

е-mail: gialid@mail.ru

 Аннотация. Рассмотрен авторский опыт использования математической системы Maple и других компьютерных инструментов для создания научно-популярных проектов физико-математического направления в рамках школьного научного общества.

На протяжении более 10 лет наша школа наряду с различными информационными технологиями работает с системой компьютерной математики Maple. Один из аспектов этой деятельности  –  научное общество учащихся «ГЕОДРОМчик», научным руководителем которого является учитель математики Гибадуллина А.И. Направления деятельности ученического научного общества – знакомство с пакетом Maple; освоение компьютерных инструментов, позволяющих работать с графикой, видео, создавать интерактивные меню; работа над индивидуальными научно-популярными проектами и создание авторских тематических электронных журналов, содержащих элементы научного исследования и математического моделирования. Компьютерная математика находит все более широкое применение – от научных исследований до продукции масскультур. Математическое моделирование проникло и в сферу создания рисунка, и в киноиндустрию. Изучение и использование учащимися нашего школьного общества символьных систем, в частности Maple, – это попытка приобщиться к современной мировой культуре компьютерного математического моделирования.

В данной статье описывается личный опыт автора, как одного из членов школьного НОУ.  

Знакомство с математической системой Maple началось с работы над проектом «Построение анимированной математической 3D-модели открывающейся книги» в 6-ом классе. Этот проект представляет собой создание пространственного анимированного изображения открывающейся книги средствами аналитической геометрии. В среде Maple была построена поэтапная программа получения этого изображения (таблицы 1 и 2). 

Таблица 1. Фрагмент программы получения анимированного изображения. 

 Рис. 1. Кадры анимации книги

Следующий проект, выполненный в среде Maple совместно с Нигометзяновой Эльзой в 7-ом классе, – короткометражный мультфильм «Колобок в лесу».

a)      b)   

 Рис. 2. Кадры анимации мультфильма

В 8-ом классе велась работа по техническому переводу сайта компании Waterloo Maple Inc. [3]. Как известно, такой перевод имеет свои особенности, которые не предусмотрены в школьной программе по изучению английского языка, поэтому опыт такой работы способствует совершенствованию владения английским языком.

В 9-ом классе началась работа над электронным журналом по космологии «Вселенная: теория и факты». Черные дыры Вселенной – один из самых загадочных и любопытных для человека объектов. Их изучение привело к интересу к астрофизике вообще. Знакомство с понятием черной дыры неизбежно вынудило изучать строение Вселенной и ее геометрии [9, 10, 11, 12]. Пришлось осмысливать сложнейшие фундаментальные понятия, теории, а также элементы высшей математики [1, 5, 6, 7, 8]. Чтобы хотя бы попытаться понять огромный объем, казалось бы, беспорядочной информации, нужно было ее анализировать и систематизировать. И тогда возникла идея проекта – авторского электронного журнала. Тем более складывается парадоксальная ситуация: астрофизика бурно развивается, проникая практически во все сферы нашей жизни, а предмета астрономии в школе нет. Поэтому такой проект мог бы восполнить этот досадный пробел и помочь школьникам – и не только – в познании Вселенной. Журнал имеет следующие разделы: Вселенная, черные дыры, белые дыры, глоссарий, теории, неевклидовы геометрии, видео-опыты, интересные факты, ссылки, использованные ресурсы. Один из разделов журнала составляют Maple-разработки, в частности, визуализированная модель искривления пространства.

Далее приводится Maple-программа (табл. 2) построения визуализации деформации плоскости под шаром определенного размера. Используются библиотеки <plots> и <plottools> пакета.

 Таблица 2.  Maple–программа визуализации деформации плоскости. 

 При h:=1:  k:=1:

 1)      2)      3)   

4)      5)      6)   

При h:=5:  k:=1:

7)      8)      9)   

Рис. 3. Кадры анимации при заданных параметрах.

Долго подбиралась функция глубины "ямы". Наконец, была найдена – это стало понятно после просмотра лекции А.Линде, где говорится об экспоненциальных процессах [13].

Меняя только параметры h и k (задающие размеры шара и ширины «ямы») и прокручивая программу снова, меняется и визуализация. Надо заметить, что построена всего лишь математическая модель визуализации, а не самого процесса.

Этот раздел предполагается пополнять новыми разработками, выполненными в среде Maple.

Журнал имеет удобную систему ссылок и организован так, что его можно оперативно обновлять. Астрофизика бурно развивается, поэтому журнал не потеряет своей актуальности.

 Заключение.

В течение 4-х лет занятий в научном обществе авторские проекты были представлены на различных сайтах, конкурсах, конференциях, форумах федерального и международного уровней:

• сайт еxponenta.ru в разделе студенческих работ [4];
• Конкурс исследовательских и творческих работ «Нобелевские надежды КНИТУ»
• Республиканский конкурс «Арт-дебют»
• V Международная ассамблея школьников (участие и публикация) [2]
• Всероссийский Горчаковский форум в г.Санкт-Петербург
• Поволжская научной конференция учащихся им. Н.И.Лобачевского
• Всероссийский фестиваль «Нескучная наука» в г.Санкт-Петербург
• Пост н.р. Гибадуллиной А.И. на сайте компании Maplesoft  http://www.mapleprimes.com/users/Alsu

  Использованная литература

 [1] Матросов А.В. Maple 6: Решение задач высшей математики и механики: Практическое руководство. – СПб.: БХВ – Петербург, 2001 г. – 528 с.

[2] V Международная Интеллектуальная Ассамблея школьников: сборник научно-исследовательских работ / Отв. ред. М. В. Волкова – Чебоксары: НИИ педагогики и психологии, 2012 – 136с. (с. 44–45)

[3] Сайт компании Maplesoft. – Режим доступа:  http://www.maplesoft.com

[4] Сайт <exponenta.ru> / Архив студенческих работ – Режим доступа:

http://www.exponenta.ru/educat/referat/XXIVkonkurs/5/index.asp

[5] Высшая математика: Учеб. Пособие для студентов пед. ин-тов по спец. 2120 «Общетехн. дисциплины и труд» / Г. Луканкин, Н. Мартынов, Г. Шадрин, Г. Яковлев; Под. ред. Г.Н. Яковлева. – М.: Просвещение, 1988. – 431 с.: ил.

[6] Справочник по высшей математике / М. Я. Выгодский. – М.: ООО «Издательство Астрель»: ООО «Издательство АСТ», 2002. – 992 с.: ил.

[7] Математический словарь высшей школы: Общ. часть/В. Т. Воднев, А. Ф. Наумович, Н.Ф. Наумович; Под ред. Ю.С. Богданова. – 2-е изд. – М.:Изд-во МПИ, 1988 – 527 с., ил.

[8] Толковый математический словарь. Основные термины: около 2500 терминов. – М.: Рус. яз., 1989. – 244 с., 186 ил.

[9] Открываем неевклидову геометрию. Кн. для внеклас. чтения учащихся 9-10 кл. сред. шк. – М.: Просвещение, 1988. – 126 с.: ил. – (Мир Знаний).

[10] Геометрия: Учебник для вузов. – СПб.: Издательство «Лань», 2003. – 416 с., ил. – (Учебники для вузов. Специальная литература)

[11] Основания геометрии: Учебн. пособие для вузов. – М.: Наука. Гл. ред. физ.-мат. лит., 1987. – 288 с.

[12] Обзорные лекции по геометрии к государственному экзамену по математике, Х семестр, курс лекций с примерами решений задач (в помощь выпускнику), проф. Ю.Г. Игнатьева. Программный продукт BIBLIO профессора Ю.Г. Игнатьева, Казань 2002 г.

[13] Видеозапись лекции Андрея Дмитриевича Линде, Стэнфордский университет (США), профессор «Многоликая Вселенная», прямая ссылка: http://elementy.ru/lib/430484