MATH4221 Euclidean and Non-Euclidean Geometries (2012 Spring) © Henry Cheng @ HKUST

Tutorial Note 4 Feb 29, 2012 (Week 5)

MATH4221 Euclidean and Non-Euclidean Geometries Tutorial Note 4 Topics covered in week 4: 7. Projective Geometry and Affine Geometry 7. Projective Geometry and Affine Geometry What you need to know:  Definition of projective geometry, an equivalent formulation  Projective completion of affine planes 

Duality of projective geometries

Last week we have looked at incidence geometries with various parallel properties.

In

particular, an affine geometry is an incidence geometry having the Euclidean parallel property. This week we focus on projective geometry, which is an incidence geometry having the elliptic parallel property and every line is incident to at least three points. In other words, to show that ℳ is a projective geometry, we need to show that the following four axioms hold in ℳ: (I1) For every two distinct points, there exists a unique line incident to both of them. (I2’) Every line is incident to at least three distinct points. (I3) There exist three points that are not collinear. (E) There are no parallel lines. The smallest projective geometry (“the smallest” refers to the number of points here) is the Fano plane, which consists of 7 points and 7 lines, and each line is incident to 3 points. We have looked at this model last week, in Example 6.1. The diagram on the right illustrates the Fano plane. 𝑙

𝜋

𝑃

𝜎 𝑐

𝑆2

Fano plane

We have also looked at some projective geometries containing infinitely many points already. Some examples include the two isomorphic models ℳ𝐸 in

𝑚

𝑃′ “points”: lines through origin

“points”: pairs of antipodal points

“lines”: planes through origin

“lines”: great circles

Page 1 of 13

Example 4.3 (b) and ℙ2 in Example 5.1 (b). Take a look at the axiom checkings again, in case you have missed them.

MATH4221 Euclidean and Non-Euclidean Geometries (2012 Spring) © Henry Cheng @ HKUST

Tutorial Note 4 Feb 29, 2012 (Week 5)

Some people would prefer defining a projective geometry as a model satisfying the following three axioms instead of our four axioms above (though we are not going to use this definition): (I1) For every two distinct points, there exists a unique line incident to both of them. (E) There are no parallel lines. These four points are said (Q) There exists four points, no three of which are collinear. to form a quadrangle. It can be shown that this definition is in fact equivalent to ours. Example 7.1

(Greenberg 2.M7)

Show that a model satisfies the axioms (I1), (I2’), (I3) and (E)

if and only if it satisfies the axioms (I1), (E) and (Q). Proof: To prove the “⇒” part, it suffices to show that (Q) holds under the assumptions that the four axioms (I1), (I2’), (I3) and (E) all hold: From (I3), there exist three points 𝑃, 𝐴 and 𝐵 which are non-collinear. Now let 𝑙 be line incident to both 𝑃 and 𝐴, and 𝑚 be the line incident

𝑃

𝐴

𝐶

𝑙

to both 𝑃 and 𝐵. It follows that 𝑙 and 𝑚 are two distinct lines. 𝐵 𝐷 𝑚 Since each line is incident to at least three distinct points by (I2’), there exist 𝐶 incident to 𝑙 and 𝐷 incident to 𝑚, such that 𝑃, 𝐴, 𝐵, 𝐶 and 𝐷 are all distinct. It follows that 𝐴, 𝐵, 𝐶 and 𝐷 are four points such that no three of them are collinear, because if any three of them were collinear, then Axiom (I1) would imply that 𝑃, 𝐴 and 𝐵 are collinear, which gives a contradiction. To prove the “⇐” part, we need to show that both (I2’) and (I3) hold under the assumptions that the three axioms (I1), (E) and (Q) all hold: (I2’) From (Q), there exist four points 𝐴, 𝐵, 𝐶 and 𝐷, no three of which are collinear. Now given a line 𝑙, we consider the following two cases. (i) If 𝑙 is one of the six lines incident to two of the points 𝐴, 𝐵, 𝐶 and 𝐷, 𝑙 ⃡ , without loss of generality. then we assume that 𝑙 is the line 𝐴𝐵 𝐴 ⃡ Now the line 𝐶𝐷 must meet 𝑙 by (E). This intersection point must be

𝐶

distinct from both 𝐴 and 𝐵, because otherwise (I1) would imply that either 𝐴, 𝐶, 𝐷 are collinear or 𝐵, 𝐶, 𝐷 are collinear, a contradiction. (ii) If 𝑙 is not one of those six lines, then at least three of 𝐴, 𝐵, 𝐶 and 𝐷 are not incident to 𝑙. We assume that 𝐴, 𝐵 and 𝐶 are

𝐶 𝐴

not incident to 𝑙, without loss of generality. ⃡ and ⃡𝐴𝐶 must Now by (E), each of the three distinct lines ⃡𝐴𝐵 , 𝐵𝐶 𝑙 meet 𝑙. These three intersection points must be distinct, because otherwise 𝐴, 𝐵 and 𝐶 would be collinear by (I1), which gives a contradiction. So in both cases, 𝑙 is incident to at least three points. (I3) This follows immediately from (Q). Page 2 of 13

𝐷

𝐵

𝐵

∎

MATH4221 Euclidean and Non-Euclidean Geometries (2012 Spring) © Henry Cheng @ HKUST

Tutorial Note 4 Feb 29, 2012 (Week 5)

In Assignment 1, you were asked to show that all lines in a projective plane are incident to the same number of points. In fact the same property holds for affine planes as well. Example 7.2

(Greenberg 2.14d, M4) Let 𝒜 be a finite affine plane. (a) Show that all lines in 𝒜 are incident to the same number of points. (b) Denote the number of points in (a) as 𝑛. (Of course 𝑛 ≥ 2 by axiom (I2).) Now show that (i) Each point in 𝒜 is incident to exactly (𝑛 + 1) lines; (ii) There are altogether 𝑛2 points in 𝒜; and (iii) There are altogether 𝑛2 + 𝑛 lines in 𝒜.

Proof: (a) Let 𝑙1 and 𝑙2 be any two distinct lines in 𝒜. We consider the following two cases: (i) If every point in 𝒜 is incident to either 𝑙1 or 𝑙2 , then 𝑙1 must be parallel to 𝑙2 by the Euclidean parallel property. We will show that both 𝑙1 and 𝑙2 are incident to two points only: Suppose not, then we assume, without loss of generality, that 𝑙1 is incident to three distinct points, say 𝑃, 𝑄 and 𝑅. Also let 𝐴 and 𝐵 be distinct points incident to 𝑙2 . ⃡ , giving a Then ⃡𝐴𝑃 and ⃡𝐴𝑄 are two distinct lines incident to 𝐴 parallel to 𝐵𝑅 contradiction to the Euclidean parallel property. (ii) If there exists a point 𝑃 in 𝒜 incident to neither 𝑙1 nor 𝑙2 , then we let 𝜈𝑃 be the number of lines 𝑃 is incident to. Then by the Euclidean parallel property, both 𝑙1 and 𝑙2 are incident to exactly (𝜈𝑃 − 1) points. So in both cases, 𝑙1 and 𝑙2 are incident to the same number of points.

∎

(b) (i) From (a), we may denote 𝑛 as the number of points that each line in 𝒜 is incident to. Let 𝑃 be any point in 𝒜 and 𝑙 be any line in 𝒜 not incident to 𝑃. Since there are 𝑛 points incident to 𝑙, we denote them as 𝑃1 , 𝑃2 , … , 𝑃𝑛 . Note that 𝑃 𝑚0 𝑚 1  There is exactly one line 𝑚0 incident to 𝑃 parallel to 𝑙; 𝑚2 𝑚𝑛  The lines 𝑚𝑖 incident to 𝑃 and 𝑃𝑖 , where 1 ≤ 𝑖 ≤ 𝑛, 𝑃1 𝑃2 are all distinct since 𝑃, 𝑃1 , 𝑃2 , … , 𝑃𝑛 are distinct points; and 𝑃𝑛 𝑙  Every line incident to 𝑃 not parallel to 𝑙 meets 𝑙 at one of 𝑃1 , 𝑃2 , … , 𝑃𝑛 . So 𝑚0 , 𝑚1 , 𝑚2 , … , 𝑚𝑛 are the only (𝑛 + 1) lines incident to 𝑃. ∎ (ii) Fix a point 𝑃 in 𝒜. We note that  𝑃 is incident to (𝑛 + 1) distinct lines;  Each of these (𝑛 + 1) lines are incident to (𝑛 − 1) points apart from 𝑃;  All these (𝑛 + 1)(𝑛 − 1) points are distinct, because otherwise (I1) is violated. Therefore there are altogether (𝑛 + 1)(𝑛 − 1) + 1 = 𝑛2 points in 𝒜. Page 3 of 13

∎

MATH4221 Euclidean and Non-Euclidean Geometries (2012 Spring) © Henry Cheng @ HKUST

Tutorial Note 4 Feb 29, 2012 (Week 5)

(iii) We first claim that any line 𝑙 in 𝒜 is parallel to (𝑛 − 1) distinct lines. To show this, we let 𝑚 be any line meeting 𝑙 at one point 𝑃. Notice that each line parallel to 𝒍 must meet 𝒎, otherwise the Euclidean parallel property is violated. Now each of the (𝑛 − 1) points on 𝑚, except 𝑃, is incident to a unique line parallel to 𝑙, by the Euclidean parallel property. So there are (𝑛 − 1) distinct lines in 𝒜 parallel to 𝑙. Now fix a line 𝑙 in 𝒜. We note that  𝑙 is incident to 𝑛 distinct points;  Each of these 𝑛 points are incident to 𝑛 distinct lines apart from 𝑙;  All these 𝑛2 lines are distinct, because otherwise (I1) is violated; and  𝑙 is parallel to (𝑛 − 1) distinct lines. Therefore there are altogether 𝑛2 + (𝑛 − 1) + 1 = 𝑛2 + 𝑛 lines in 𝒜.

∎

There is a nice connection between affine geometries and projective geometries. By adjoining points at infinity (equivalence classes of equal or parallel lines) and a line at infinity (set of all points at infinity) to an affine plane, we obtain its projective completion, which is a projective plane. Conversely, any projective plane is the projective completion of some affine plane. Example 7.3

(Greenberg 2.M2)

Show that every projective plane is isomorphic to the

projective completion of some affine plane. Proof: Given any projective plane 𝒫, we remove from 𝒫 a line 𝑙0 and all the points incident to 𝑙0 . We denote the remaining model as 𝒜, and we aim to show that 𝒜 is an affine plane. (I1) Since every two points in 𝒫 determine a unique line and all the points incident to 𝑙0 are removed, it is clear that every two points in 𝒜 determine a unique line. (I2) Since each line in 𝒫 is incident to at least three points and 𝑙0 has exactly one intersection point with each line in 𝒜, each line in 𝒜 is incident to at least two points. (I3) From Example 7.1 we see that there exist four points 𝐴, 𝐵, 𝐶 and 𝐷 in 𝒫, no three of which are collinear. We consider the following two cases: (i) If 𝑙0 is not one of the six lines incident to two of the points 𝐴, 𝐵, 𝐶 and 𝐷, then at least three of 𝐴, 𝐵, 𝐶 and 𝐷 remains in 𝒜, and they are not collinear. (ii) If 𝑙0 is one of those six lines, say the line incident to both 𝐴 and 𝐵, then since ⃡𝐴𝐶 ⃡ and 𝐵𝐷 are distinct lines, they meet at some point 𝑃 which is not incident to 𝑙0 , and 𝐶, 𝐷 and 𝑃 are three non-collinear points in 𝒜. In both cases, there exists three non-collinear points in 𝒜.

Page 4 of 13

MATH4221 Euclidean and Non-Euclidean Geometries (2012 Spring) © Henry Cheng @ HKUST

Tutorial Note 4 Feb 29, 2012 (Week 5)

(A) Given any line 𝑚 in 𝒜 and any point 𝑃 in 𝒜 not incident to 𝑚, ′

𝑄

𝑃

𝑚0

the corresponding line 𝑚 in 𝒫 intersects 𝑙0 at a unique point 𝑄 𝑚 𝑙0 by the elliptic parallel property of 𝒫. Now the distinct points 𝑃 and 𝑄 determine a unique line 𝑚0 ′ in 𝒫 distinct from 𝑚′ , and the line 𝑚0 ≔ 𝑚0 ′ \{𝑄} in 𝒜 is parallel to 𝑚, otherwise (I1) for 𝒫 is violated. Moreover, the line 𝑚0 is unique. (Why?) Therefore 𝒜 is an affine plane. Now it remains to show that the projective completion 𝒜∗ of 𝒜 is isomorphic to 𝒫. We fix a point 𝑃0 in 𝒜, and define a function 𝐹: {points in 𝒫} → {points in 𝒜∗ } by 𝑃 if 𝑃 is not incident to 𝑙0 𝐹(𝑃) ≔ {the point at infinity [𝑙𝑃 ], where , if 𝑃 is incident to 𝑙0 ⃡ 𝑙𝑃 is the line 𝑃𝑃0 \𝑙0 in 𝒜 and define a function 𝑓: {lines in 𝒫} → {lines in 𝒜∗ } by 𝑓(𝑙) ≔ {

(𝑙\𝑙0 ) ∪ {[𝑙\𝑙0 ]} the line at infinity 𝑙∞

if 𝑙 ≠ 𝑙0 . if 𝑙 = 𝑙0

It is quite clear that both 𝐹 and 𝑓 are bijective. (Exercise Problem 1) Moreover, if 𝑃 is incident to a line 𝑙 in 𝒫, then we consider the following two cases: (i) If 𝑃 is not incident to 𝑙0 , then 𝑙 ≠ 𝑙0 . So 𝑃 is incident to 𝑙\𝑙0 in 𝒜, which implies that 𝐹(𝑃) = 𝑃 is incident to 𝑓(𝑙) = (𝑙\𝑙0 ) ∪ {[𝑙\𝑙0 ]} in 𝒜∗ . (ii) If 𝑃 is incident to 𝑙0 , then 𝐹(𝑃) = [𝑙𝑃 ].  If 𝑙 = 𝑙0 , then the point at infinity 𝐹(𝑃) = [𝑙𝑃 ] is obviously incident to the line at infinity 𝑓(𝑙) = 𝑙∞ ;  If 𝑙 ≠ 𝑙0 , then 𝑙\𝑙0 = 𝑙\{𝑃} is equal or parallel to 𝑙𝑃 . So [𝑙𝑃 ] = [𝑙\𝑙0 ], which implies that 𝐹(𝑃) = [𝑙𝑃 ] is incident to 𝑓(𝑙) = (𝑙\𝑙0 ) ∪ {[𝑙\𝑙0 ]}. Now 𝐹 and 𝑓 are bijections that preserve incidence relations, so 𝒜∗ is isomorphic to 𝒫. ∎ Pappus’ Statement and Desargues’ Statement are two statements that hold in some special types of projective planes. In particular, they hold in the real projective plane ℙ2 . Pappus’ Statement: If 𝑙 and 𝑙 ′ are two distinct lines and 𝐴, 𝐵, 𝐶, 𝐴′ , 𝐵 ′ and 𝐶 ′ are six distinct points such that 𝐴, 𝐵 and 𝐶 are incident to 𝑙, 𝐴′ , 𝐵 ′ and 𝐶 ′ are incident to 𝑙 ′ , but none of them are incident to both 𝑙 and 𝑙 ′ , ⃡ ′ and then the three intersection points 𝑃 of ⃡𝐴𝐵 ′ and ⃡𝐴′ 𝐵 , 𝑄 of 𝐴𝐶 ⃡ ′ 𝐶 , and 𝑅 of 𝐵𝐶 ⃡ ′ and 𝐵 ⃡ ′ 𝐶 , are collinear. 𝐴 Desargues’ Statement: For any seven distinct points 𝑂, 𝐴, 𝐵, 𝐶, 𝐴′ , 𝐵 ′ and 𝐶 ′ , if {𝑂, 𝐴, 𝐴′ }, {𝑂, 𝐵, 𝐵 ′ } and {𝑂, 𝐶, 𝐶 ′ } are collinear, but {𝐴, 𝐵, 𝐶} and {𝐴′ , 𝐵 ′ , 𝐶 ′ } are not collinear, then the three intersection points 𝑃 of ⃡𝐴𝐵 and ⃡ ′ 𝐵 ′, 𝑄 of 𝐴𝐶 ⃡ and 𝐴 ⃡ ′ 𝐶 ′ , and 𝑅 of 𝐵𝐶 ⃡ and 𝐵 ⃡ ′ 𝐶 ′ , are collinear. 𝐴 Page 5 of 13

MATH4221 Euclidean and Non-Euclidean Geometries (2012 Spring) © Henry Cheng @ HKUST

Tutorial Note 4 Feb 29, 2012 (Week 5)

𝐶

𝐵

𝑂 𝐴

𝐴 𝑙 𝑙′

𝑃

𝑅 𝑄 𝐴′

𝐴′

𝐵′

𝐶 𝑄 𝐶′

𝑅

𝐶′ Desargues’ Statement

Pappus’ Statement

Example 7.4

𝐵 𝐵′

𝑃

Show that in a projective plane, Pappus’ Statement implies Desargues’ Statement.

Proof: In a projective plane 𝒫, we suppose that Pappus’ Statement holds. To show that Desargues’ Statement also holds, we let 𝑂, 𝐴, 𝐵, 𝐶, 𝐴′ , 𝐵 ′ and 𝐶 ′ be seven distinct points such that {𝑂, 𝐴, 𝐴′ }, {𝑂, 𝐵, 𝐵 ′ } and {𝑂, 𝐶, 𝐶 ′ } are collinear but {𝐴, 𝐵, 𝐶} and ⃡ {𝐴′ , 𝐵 ′ , 𝐶 ′ } are not collinear. We let 𝑃, 𝑄 and 𝑅 be the three intersection points of 𝐴𝐵 ⃡ and ⃡𝐵 ′ 𝐶 ′ respectively, and aim to show that they are and ⃡𝐴′ 𝐵 ′, of ⃡𝐴𝐶 and ⃡𝐴′ 𝐶 ′ , and of 𝐵𝐶 collinear. 𝑇

𝑈

𝐴

𝑂 𝐵 𝐶 𝑅

𝑃 𝑆

𝐴′

𝑄

𝐶′ 𝐵′

𝑉

(i) Let 𝑆 be the intersection point of ⃡𝐴𝐵 and ⃡𝐴′ 𝐶 ′ . Then the sets {𝑂, 𝐶, 𝐶 ′ } and {𝐵, 𝑆, 𝐴} are distinct points on two distinct lines. So by Pappus’ Statement, the intersection points ⃡ , 𝑈 of ⃡𝑂𝐴 and ⃡𝐵𝐶 ′ , and 𝑄 of ⃡𝐴𝐶 and ⃡𝑆𝐶 ′ , are collinear. 𝑇 of ⃡𝑂𝑆 and 𝐵𝐶 (ii) Next, the points {𝑂, 𝐵, 𝐵 ′ } and {𝐶 ′ , 𝐴′ , 𝑆} are distinct points 𝐴′, 𝐶 ′ and 𝑆 are collinear. on two distinct lines, so by Pappus’ Statement, the intersection ⃡ ′ , 𝑉 of 𝑂𝑆 ⃡ and 𝐵 ⃡ ′ 𝐶 ′ , and 𝑃 of 𝐵𝑆 ⃡ and ⃡𝐴′ 𝐵 ′, are collinear. points 𝑈 of ⃡𝑂𝐴′ and 𝐵𝐶 𝑂, 𝐴 and 𝐴′ are collinear.

𝐴, 𝐵 and 𝑆 are collinear.

Finally, the points {𝐵, 𝐶 ′ , 𝑈} and {𝑉, 𝑇, 𝑆} are distinct points on two distinct lines, so by ⃡ and 𝑈𝑉 ⃡ and ⃡𝑉𝐶 ′ , 𝑃 of 𝐵𝑆 ⃡ (by (ii)), Pappus’ Statement, the intersection points 𝑅 of 𝐵𝑇 and 𝑄 of ⃡𝑆𝐶 ′ and ⃡𝑇𝑈 (by (i)), are collinear. ∎ Remark:

The converse is not true. Desargues’ Statement does not imply Pappus’ Statement. Page 6 of 13

MATH4221 Euclidean and Non-Euclidean Geometries (2012 Spring) © Henry Cheng @ HKUST

Tutorial Note 4 Feb 29, 2012 (Week 5)

Projective planes satisfy the Principle of Duality. This means that (i)

The dual of a projective plane is still a projective plane. (Recall that this statement is not true if “projective plane” is replaced by “incidence geometry”!) (ii) If a statement is true for all projective planes, then the “dual statement” is also true for all projective planes. Example 7.5

State the dual of the following statements: (a) There exist four points, no three of which are collinear. (In other words, a quadrangle exists.) (b) (Desargues’ Statement) For any seven distinct points 𝑂, 𝐴, 𝐵, 𝐶, 𝐴′ , 𝐵 ′ and 𝐶 ′ , if {𝑂, 𝐴, 𝐴′ }, {𝑂, 𝐵, 𝐵 ′ } and {𝑂, 𝐶, 𝐶 ′ } are collinear, but {𝐴, 𝐵, 𝐶} and {𝐴′ , 𝐵 ′ , 𝐶 ′ } are not collinear, then the three intersection points 𝑃 of ⃡ ⃡ ⃡ ′ 𝐶 ′ , and 𝑅 of 𝐵𝐶 ⃡ ⃡ ′ 𝐶 ′ , are 𝐴𝐵 and ⃡𝐴′ 𝐵 ′ , 𝑄 of 𝐴𝐶 and 𝐴 and 𝐵 collinear.

Solution: (a) There exists four lines, no three of which are concurrent. (b) For any seven distinct lines 𝑜, 𝑙1, 𝑙2 , 𝑙3 , 𝑚1 , 𝑚2 and 𝑚3 , if {𝑜, 𝑙1 , 𝑚1 }, {𝑜, 𝑙2 , 𝑚2 } and {𝑜, 𝑙3 , 𝑚3 } are concurrent, but {𝑙1 , 𝑙2 , 𝑙3 } and {𝑚1 , 𝑚2 , 𝑚3 } are not concurrent, then the three lines 𝑛1 determined by the intersection points of {𝑙1 , 𝑙2 } and of {𝑚1 , 𝑚2 }, 𝑛2 determined by the intersection points of {𝑙1 , 𝑙3 } and of {𝑚1 , 𝑚3 }, and 𝑛3 determined by the intersection points of {𝑙2 , 𝑙3 } and {𝑚2 , 𝑚3 }, are concurrent. In the above example, since the statement in (a) is true for all projective planes, its dual statement is also true for all projective planes, by the Principle of Duality. Desargues’ Statement is not always true, though it actually imply its dual statement. The same applies to Pappus’ Statement (Exercise Problems 4-5). Example 7.6

In a projective plane, show that Desargues’ Statement implies its dual statement.

Proof: Suppose that Desargues’ Statement holds in a projective plane 𝒫. To show that the dual statement also holds, we let 𝑜, 𝑙1, 𝑙2 , 𝑙3 , 𝑚1 , 𝑚2 and 𝑚3 be seven distinct lines such that {𝑜, 𝑙1 , 𝑚1 }, {𝑜, 𝑙2 , 𝑚2 } and {𝑜, 𝑙3 , 𝑚3 } are concurrent but {𝑙1 , 𝑙2 , 𝑙3 } and {𝑚1 , 𝑚2 , 𝑚3 } are not concurrent. We let 𝑛1 be the line determined by the intersection points of {𝑙1 , 𝑙2 } and of {𝑚1 , 𝑚2 }, 𝑛2 be the line determined by the intersection points of {𝑙1 , 𝑙3 } and of {𝑚1 , 𝑚3 }, and 𝑛3 be the line determined by the intersection points of {𝑙2 , 𝑙3 } and {𝑚2 , 𝑚3 }, and aim to show that 𝑛1 , 𝑛2 and 𝑛3 are concurrent. Page 7 of 13

MATH4221 Euclidean and Non-Euclidean Geometries (2012 Spring) © Henry Cheng @ HKUST

Tutorial Note 4 Feb 29, 2012 (Week 5)

Now we give labels to the intersection points. Denote 𝑂, 𝐴, 𝐴′ , 𝐵, 𝐵 ′ , 𝐶 and 𝐶 ′ as the intersection points of {𝑜, 𝑙1 , 𝑚1 } , {𝑜, 𝑙2 , 𝑚2 } , {𝑜, 𝑙3 , 𝑚3 } , {𝑙1 , 𝑙2 } , {𝑙1 , 𝑙3 } , {𝑚1 , 𝑚2 } and {𝑚1 , 𝑚3 } respectively. Then we can easily see that (i) {𝑂, 𝐴, 𝐴′ } , {𝑂, 𝐵, 𝐵 ′ } and {𝑂, 𝐶, 𝐶 ′ } are collinear (on the three lines 𝑜 , 𝑙1 and 𝑚1 respectively). (ii) {𝐴, 𝐵, 𝐶} and {𝐴′ , 𝐵 ′ , 𝐶 ′ } are not collinear (otherwise some of the seven lines will be the same, which is a contradiction). Therefore by Desargues’ Statement, the three intersection points 𝑃 of ⃡𝐴𝐵 = 𝑙2 and ⃡ ′ 𝐵 ′ = 𝑙3 , 𝑄 of 𝐴𝐶 ⃡ = 𝑚2 and 𝐴 ⃡ ′ 𝐶 ′ = 𝑚3 , and 𝑅 of 𝐵𝐶 ⃡ = 𝑛1 and 𝐵 ⃡ ′ 𝐶 ′ = 𝑛2 , are 𝐴 collinear. ⃡ is just 𝑛3 . But the line 𝑃𝑄

Therefore 𝑛1 , 𝑛2 and 𝑛3 are concurrent at 𝑅.

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MATH4221 Euclidean and Non-Euclidean Geometries (2012 Spring) © Henry Cheng @ HKUST

Tutorial Note 4 Feb 29, 2012 (Week 5)

Exercise 1. Complete the proof in Example 7.3 by showing that both the functions 𝐹 and 𝑓 are bijective. 2. (Bennett 2.1.8, 2.2.2) Let 𝒜 be an affine plane. (a) Show that for every point 𝑃 in 𝒜, 𝑃 is incident to at least three distinct lines. (b) Define a relation ~ on the set of lines in 𝒜 by 𝑙 ~ 𝑚 ⇔ (𝑙 = 𝑚 or 𝑙 is parallel to 𝑚). Show that ~ is an equivalence relation. (c) Show that there exist at least three distinct ~-equivalence classes in the set of lines in 𝒜. 3. Show that any two affine planes with 9 distinct points are isomorphic. 4. State the dual of the following statements: (a) (Pappus’ Statement) If 𝑙 and 𝑙 ′ are two distinct lines and 𝐴, 𝐵, 𝐶, 𝐴′ , 𝐵 ′ and 𝐶 ′ are six distinct points such that 𝐴, 𝐵 and 𝐶 are incident to 𝑙, 𝐴′ , 𝐵 ′ and 𝐶 ′ are incident to 𝑙 ′ , but none of them are incident to both 𝑙 and 𝑙 ′ , then the three intersection points 𝑃 of ⃡𝐴𝐵 ′ and 𝐴 ⃡ ′ 𝐵, 𝑄 of ⃡𝐴𝐶 ′ and ⃡𝐴′ 𝐶 , and 𝑅 of ⃡𝐵𝐶 ′ and ⃡𝐵 ′ 𝐶 , are collinear. (b) (Fano’s Statement) If 𝐴, 𝐵, 𝐶 and 𝐷 are four points, no three of which are collinear, then the three intersection points 𝑃 of ⃡𝐴𝐵 and ⃡𝐶𝐷, 𝑄 of ⃡𝐴𝐶 and ⃡𝐵𝐷, and 𝑅 of ⃡𝐴𝐷 ⃡ , are not collinear. and 𝐵𝐶 5. Recall Pappus’ Statement, Fano’s Statement and their respective dual statements in Problem 4. Show that in a projective plane, (a) Pappus’ Statement implies its dual statement; and (b) Fano’s Statement implies its dual statement. 6. Determine whether each of the following statements holds in the Fano plane (i.e. the projective plane with seven points): (a) Pappus’ Statement (b) Desargues’ Statement (c) Fano’s Statement 7. Suppose that we are given a poker deck of 52 cards (without the jokers). We do not distinguish the four suits. Arrange the 52 cards into 13 stacks of four cards each, such that all of the following conditions are satisfied: (i) For any two different ranks (A, 2, 3, 4 and so on), there exists exactly one stack containing both of these two ranks. (ii) For any two different stacks, there exists exactly one rank such that each of these two stacks contains this rank. (iii) There exist four different ranks such that any three of them do not appear simultaneously in any single stack. Give your answer by listing the 13 stacks as multisets, e.g. 𝒮 = {{1234}, {2256}, {346K}, … }. Hint:

Try to relate the above three conditions to some axioms we have seen before.

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