Euclidean Geometry

I Fill in the right terms: plane, expression, solid solve generalize restriction applying.

Euclidean geometry is the study of ___ and ___ figures on the basis of axioms and theorems employed by the Greek mathematician Euclid (c. 300 BC). Indeed, until the second half of the 19th century, when non-Euclidean geometries attracted the attention of mathematicians, geometry meant Euclidean geometry. It is the most typical 3_____ of general mathematical thinking. Rather than the memorization of simple algorithms to 4 ____ equations by rote, it demands true insight into the subject, clever ideas for 5 ____ theorems in special situations, an ability to 6 _____ from known facts, and an insistence on the importance of proof. In Euclid's great work, the Elements, the only tools employed for geometrical constructions were the ruler and compass—a 7 _____ retained in elementary Euclidean geometry to this day.

Euclid realized that a rigorous development of geometry must start with the foundations. Hence, he began the Elements with some undefined terms, such as “a point is that which has no part” and “a line is a length without breadth.” Proceeding from these terms, he defined further ideas such as angles, circles, triangles, and various other polygons and figures. For example, an angle was defined as the inclination of two straight lines, and a circle was a plane figure consisting of all points that have a fixed distance (radius) from a given centre. As a basis for further logical deductions, Euclid proposed five common notions, such as “things equal to the same thing are equal,” and five unprovable but intuitive principles known variously as postulates or axioms. Stated in modern terms, the axioms are as follows (match the bits, II):

1. Given two points, ...

2. A straight line segment ...

3. A circle can be constructed ...

4. All right angles...

5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, ...

1. can be prolonged indefinitely.

2. when a point for its centre and a distance for its radius are given.

3. will meet on that side on which the angles are less than the two right angles.

4. there is a straight line that joins them.

5. are equal.

All five axioms provided the basis for numerous provable statements, or theorems, on which Euclid built his geometry. Summarizing the information below, the five most important theorems of plane Euclidean geometry are: the sum of the angles in a triangle is 180 degrees, the Bridge of Asses, the fundamental theorem of similarity, the Pythagorean theorem, and the invariance of angles subtended by a chord in a circle. Most of the more advanced theorems of plane Euclidean geometry are proven with the help of these theorems.

2. IV Match the title with the corresponding paragraph: The Similarity of Triangles, Circles, the Congruency of Triangles, Areas, the Pythagorean Theorem.

A ______. Two triangles are said to be congruent if one can be exactly superimposed on the other by a rigid motion, and the congruence theorems specify the conditions under which this can occur. The first theorem illustrated in the diagram is the side-angle-side (SAS) theorem: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. Following this, there are corresponding angle-side-angle (ASA) and side-side-side (SSS) theorems.

The first very useful theorem derived from the axioms is the basic symmetry property of isosceles triangles—i.e., that two sides of a triangle are equal if and only if the angles opposite them are equal. Euclid's proof of this theorem was once called Pons Asinorum (“Bridge of Asses”), supposedly because mediocre students could not proceed across it to the farther reaches of geometry. The Bridge of Asses opens the way to various theorems on the congruence of triangles. 

B ____________. As indicated above, congruent figures have the same shape and size. Similar figures, on the other hand, have the same shape but may differ in size. Shape is intimately related to the notion of proportion, as ancient Egyptian artisans observed long ago. Segments of lengths a, b, c, and d are said to be proportional if a:b = c:d (read, a is to b as c is to d; in older notation a:b::c:d). The fundamental theorem of similarity states that a line segment splits two sides of a triangle into proportional segments if and only if the segment is parallel to the triangle's third side. The similarity theorem may be reformulated as the AAA (angle-angle-angle) similarity theorem: two triangles have their corresponding angles equal if and only if their corresponding sides are proportional. Two similar triangles are related by a scaling (or similarity) factor s: if the first triangle has sides a, b, and c, then the second one will have sides sa, sb, and sc.

3. C ______. Just as a segment can be measured by comparing it with a unit segment, the area of a polygon or other plane figure can be measured by comparing it with a unit square. The common formulas for calculating areas reduce this kind of measurement to the measurement of certain suitable lengths. The simplest case is a rectangle with sides a and b, which has area ab. By putting a triangle into an appropriate rectangle, one can show that the area of the triangle is half the product of the length of one of its bases and its corresponding height—bh/2. One can then compute the area of a general polygon by dissecting it into triangular regions. If a triangle (or more general figure) has area A, a similar triangle (or figure) with a scaling factor of s will have an area of s2A.

4.  D _____. For a triangle ABC the Pythagorean theorem has two parts: (1) if ACB is a right angle, then a2 + b2 = c2; (2) if a2 + b2 = c2, then ACB is a right angle. For an arbitrary triangle, the Pythagorean theorem is generalized to the law of cosines: a2 + b2 = c2 − 2ab cos (ACB). When ACB is 90 degrees, this reduces to the Pythagorean theorem because cos (90°) = 0. Despite its antiquity, the theorem remains one of the most important in mathematics. It enables one to calculate distances or, more importantly, to define distances in situations far more general than elementary geometry. For example, it has been generalized to multidimensional vector spaces.

E ______. A chord AB is a segment in the interior of a circle connecting two points (A and B) on the circumference. When a chord passes through the circle's centre, it is a diameter, d. The circumference of a circle is given by πd, or 2πr where r is the radius of the circle; the area of a circle is πr2. In each case, π is the same constant (3.14159…). The Greek mathematician Archimedes (c. 285–212/211 BC) used the method of exhaustion to obtain upper and lower bounds for π by circumscribing and inscribing regular polygons about a circle.

 

A semicircle has its end points on a diameter of a circle. Thales (flourished 6th century BC) is generally credited with proving that any angle inscribed in a semicircle is a right angle; that is, for any point C on the semicircle with diameter AB, ACB will always be 90 degrees. Another important theorem states that for any chord AB in a circle, the angle subtended by any point on the same semiarc of the circle will be invariant. Slightly modified, this means that in a circle, equal chords determine equal angles, and vice versa.

TASKS

V Explain the words in bold in your own words.

VI Say whether the statements are true or false:

• In Euclidean geometry you need to memorize everything by rote.

• The use of tools in EG is restricted.

• If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are similar.

• The area of a polygon can be measured by comparing it with a unit segment.

• Thales has proved that that any angle inscribed in a semicircle is a right angle.

VII Answer the questions:

• In what way is geometry different from other mathematical disciplines?

• What effect has EG had on the present-day science?

• What are the foundations introduced by Euclid?

• What is an axiom? What makes it special?

• Why was the proof of the theorem called “The Bridge of Asses”?

• What is the difference between the similarity and congruency of triangles?

• What is the simplest case of an area?

• Apart the one given in the example, where else is the Pythagorean theorem used? (Use your background knowledge or any sources you can).

• How can you explain the difference between the inscribed and circumscribed polygon?

VIII You are giving a lecture on the basics of Euclidean Geometry. Don’t forget to mention the key figures in it, its fundamental terms and 5 most important theorems (use some extra materials to expand the information you provide). The rest of the class will play the role of the listeners (so, be ready to ask questions to get clearer explanations). Do not forget that the lecture should be interactive!

Text IV

Non-Euclidean Geometries

When Euclid presented his axiomatic treatment of geometry, one of his assumptions, his fifth postulate, appeared to be less obvious or fundamental than the others. As it is now conventionally formulated, it asserts that there is exactly one parallel to a given line through a given point. Attempts to derive this from Euclid's other axioms did not succeed, and, at the beginning of the 19th century, it was realized that Euclid's fifth postulate is, in fact, independent of the others. It was then seen that Euclid had described not the one true geometry but only one of a number of possible geometries.

A ____ Within the framework of Euclid's other four postulates (and a few that he omitted), there were also possible elliptic and hyperbolic geometries. In plane elliptic geometry there are no parallels to a given line through a given point; it may be viewed as the geometry of a spherical surface on which antipodal points have been identified and all lines are great circles. This was not viewed as revolutionary. More exciting was plane hyperbolic geometry, developed independently by the Hungarian mathematician János Bolyai (1802–60) and the Russian mathematician Nikolay Lobachevsky (1792–1856), in which there is more than one parallel to a given line through a given point. This geometry is more difficult to visualize, but a helpful model presents the hyperbolic plane as the interior of a circle, in which straight lines take the form of arcs of circles perpendicular to the circumference.

Another way to distinguish the three geometries is to look at the sum of the angles of a triangle. It is 180° in Euclidean geometry, as first reputedly discovered by Thales of Miletus (fl. 6th century BC), whereas it is more than 180° in elliptic and less than 180° in hyperbolic geometry.

Figure 2: Contrasting triangles in Euclidean, elliptic, and hyperbolic spaces.

B _____ The discovery that there is more than one geometry was of foundational significance and contradicted the German philosopher Immanuel Kant (1724–1804). Kant had argued that there is only one true geometry, Euclidean, which is known to be true a priori by an inner faculty (or intuition) of the mind. For Kant, and practically all other philosophers and mathematicians of his time, this belief in the unassailable truth of Euclidean geometry formed the foundation and justification for further explorations into the nature of reality. With the discovery of consistent non-Euclidean geometries, there was a subsequent loss of certainty and trust in this innate intuition, and this was fundamental in separating mathematics from a rigid adherence to an external sensory order (no longer vouchsafed as “true”) and led to the growing abstraction of mathematics as a self-contained universe. This divorce from geometric intuition added impetus to later efforts to rebuild assurance of truth on the basis of logic.

What then is the correct geometry for describing the space (actually space-time) we live in? It turns out to be none of the above, but a more general kind of geometry, as was first discovered by the German mathematician Bernhard Riemann (1826–66). In the early 20th century, Albert Einstein showed, in the context of his general theory of relativity, that the true geometry of space is only approximately Euclidean. It is a form of Riemannian geometry in which space and time are linked in a four-dimensional manifold, and it is the curvature at each point that is responsible for the gravitational “force” at that point. Einstein spent the last part of his life trying to extend this idea to the electromagnetic force, hoping to reduce all physics to geometry, but a successful unified field theory eluded him.

C ______ In the 19th century, the German mathematician Georg Cantor (1845–1918) returned once more to the notion of infinity and showed that, surprisingly, there is not just one kind of infinity but many kinds. In particular, while the set ℕ of natural numbers and the set of all subsets of ℕ are both infinite, the latter collection is more numerous, in a way that Cantor made precise, than the former. He proved that ℕ, ℤ, and Q all have the same size, since it is possible to put them into one-to-one correspondence with one another, but that R is bigger, having the same size as the set of all subsets of N.

However, Cantor was unable to prove the so-called continuum hypothesis, which asserts that there is no set that is larger than N yet smaller than the set of its subsets. It was shown only in the 20th century, by Gödel and the American logician Paul Cohen (b. 1934), that the continuum hypothesis can be neither proved nor disproved from the usual axioms of set theory. Cantor had his detractors, most notably the German mathematician Leopold Kronecker (1823–91), who felt that Cantor's theory was too metaphysical and that his methods were not sufficiently constructive.

Match the titles with the paragraphs: Cantor, Riemannian Geometry, Elliptic and Hyperbolic Geometries.

Explain the terms: visualize, approximately, revolutionary, precise, four-dimensional, to elude, constructive.

Say whether the statements are true or false:

• Lobachevsky first introduced elliptic geometry.

• Other geometries issue from Euclid's postulates.

• The sum of the angles of a triangle is more than 180° in hyperbolic and less than 180° in elliptic geometry.

• The innate intuition has been the basic principle for all the researchers.

• Later on logical basis became crucial.

• Cantor's theory was too metaphysical to be explained from the point of view of traditional geometry.

Answer the questions:

• What is the reason for creating a number of possible geometries?

• What can be understood by Elliptic geometry?

• In what way are elliptic and hyperbolic geometries different from plain geometry? How do we distinguish them? Describe what you can see in the picture?

• What was Immanuel Kant's view upon geometry? What is meant by the innate intuition?

• What was the reason for further explorations into the nature of reality?

• In what way is geometry linked to the Einstein's Theory of Relativity?

• What notion was brought back by Cantor in the 19^th century?

• What does the continuum hypothesis consist in? Why is it considered metaphysical?

Speaking. Speak about the researchers mentioned in the article. In what way did they contribute into the history of science? Pay special attention to the Cantor's theory.

Appendix I

Tasks for Video Lessons

Lead-in

• What is a black hole? Why is it called so?

• Why do black holes arouse so much interest among cosmologists?

Supermassive Black Holes

Part I

I Watch the episode. Make sure you can explain the following:

• What the aim of cosmology is;

• What a supermassive black hole is;

• Why supermassive black holes are supposed to be so destructive;

• What “singularity” means;

• The difference between active and inactive galaxies;

• What the term “quasar” means;

• What kind of device a spectroscope is.

II Watch the episode for the second time. Fill in the gaps:

• When the ________ was young there were no stars or planets, just the ________ _______ of ________ gas.

• We just don’t know how galaxies _______ out of the ________ hot gases that filled the Universe.

• A supermassive black hole is, quite simply, _______ gone mad, an object of such ______ _______, its ______ _______ is insatiable.

• These galaxies have a brilliant ________ _______ with vast _____ of _________ spurting out of the centre.

• A giant black hole would have a _______ ______ so _________it would haul the galaxies around it at almost the _____of _____.

• The gas _______ against ________, essentially, and gets______ ______, and ______ ______ gas burns _______.

• So, what you are looking for is the ________ of its ______.

III. Answer the questions:

1. What is the reason why a quasar shines so brightly?

2. Why can’t one actually see a black hole?

3. Why did the scientists doubt the existence of black holes for decades?

4. What exactly has been found to prove their existence?

5. Why the results obtained with the help of the spectroscope proved to be so shocking?

6. What are the 2 states of a black hole?

7. What kind of information was obtained with the help of the Hubble Space Telescope?

8. In what way is the Keck Telescope in Hawaii more efficient than Hubble? What data have been collected with its help?

9. What was the basic question cosmologists were trying to answer?

IV. Speaking. Role-play an interview with an expert in galaxies and black holes. Ask the cosmologist about the properties of black holes, the forces that govern them, the related notions and the information provided by the current research.

V. Summarize the episode in 5 – 7 sentences. Make sure you produce only relevant information!