STRUCTURAL INVARIANCE OF RIGHT-ANGLE TRIANGLE UNDER ROTATION-SIMILARITY TRANSFORMATION

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1 INTERNATIONAL JOURNAL OF GEOMETRY Vol. 7 (08), No., 66-7 STRUCTURAL INVARIANCE OF RIGHT-ANGLE TRIANGLE UNDER ROTATION-SIMILARITY TRANSFORMATION ANDREI MOLDAVANOV Abstract. We consider relations in the redecessor-successor chain emerged at rotation-similarity transformation of right-angle triangle with an arbitrary leg ratio in -dimensional Euclidean sace. Following this, we generalize structural ratios obtained earlier. We nd conditions favoring to the structural invariance for above triangle in the redecessor-successor air. Finally, we discuss image of above transformation and comare robability of folding/unfolding develoment for random transformation conditions.. Introduction In [], the family of right-angle triangles AB n C n in -dimensional Euclidean sace with leg ratio AC =BC =, where =, was considered (Figure ). It was shown that if to mar o the distance from the vertex B n along AB n as B n C n+ = B n C n = B n F n then for n, the following ratios are valid irrelevant to the sign of S n () S F n S T = n AFn ACn = = Sn T Sn C A n AC n+ = ; () (3) = S n C Sn+ F = AC n = AC (n)ex[g()b] where S n is area of AB n C n, S F n = S C n + S T n is area of AF n C n, S C n is area of AC n+ C n, S T n is area of equal triangles C n C n+ B n and C n F n B n, and is the golden ratio []. Keywords and hrases: Area invariance, Planar Transformation, Rightangle triangle, Random Develoment (00)Mathematics Subject Classi cation: 5A99, 5M5 Received: In revised form: Acceted:

2 Structural Iinvariance of Right-Angle Triangle Under Rotation-SimilarityTransformation 67 Figure. Family of right-angle triangles at lanar rotationsimilarity transformation. So, ratio () or equivalent Sn+ C =SF n = declares invariance of the referenced area ratio under above transformation. In (3), n = \C n AB n = arctan( = ), g() = for folding logarithmic siral (S n < 0 ) and g() = for unfolding one (S n > 0 ), and b = =arctan(=). In this article, we generalize (-3) for arbitrary rational = AC n =B n C n and consider some additional features for family AB n C n related to its random structuring. Write down some relations from [], we will be using further, (4) (5) (6) (7) f = + + S C n = (B nc n ) S T n = (B nc n ) S C n = (B nc n ) where f is a similarity ratio. r! + r + r! + +. Generalization theorem Formulate theorem generalizing (-3) for any and rove its statements one by one. Theorem.. If in arbitrary right-angle AB n C n with xed leg ratio AC n =B n C n =, where is any rational number, to mar o from the vertex Bn along hyotenuse AB n the line segments B n C n+ and B n F n with

3 68 Andrei Moldavanov the length equal to length of B n C n, then irrelevant to the sign of S n, for any integer n n, where = (4= ), = (8) AFn ACn = ; AC n AC n+ (9) S T n S F = () n ; S C n the full area Sn+ F (SF n ) of the rst immediate successor (redecessor) in the chain of similar triangles is exactly equal to the area Sn C (Sn+ C ) of the core triangle in its immediate redecessor (successor), i.e. (0) = : Proof. (a) Relation (5.a) immediately follows from de nition of AC n = B n C n ; AF n = AB n + B n C n = B n C n ( + + ) and AC n+ = AB n B n C n ( + ). Now, calculate () using (4 7 ) as S T () () = n S F : n = S C n S T n S T n 4f f + 4 = ; Now, rove (0 ). For seci city, consider case =, the case = - is roved by simle reversing of all the ratios. Using (4 ), () S F n = SF n f = S F n + +! then aly (7 ) which yields (3) Sn+ F = (B nc n ) r! = Sn C + that roves (0 ): 3. Distortion theorem So far, we believed that B n C n = B n C n+ and roved that at that condition, relation (0) is identity. Now, let the length of BC n+ be random, i:e: generally BC n+ 6= B n C n and evaluate the measure of distortion between S F n+ and SC n (S F n and S C n+ ). From that, random [ + ; ] with origin in the vortex B n and the full range R = AF = + + with the subrange R = for the random event S n 0 and R + = + + for the random event S n > 0. Assuming uniform robability density distribution in R, calculate aroriate robability density function (4) g () = R R = + +

4 and Structural Iinvariance of Right-Angle Triangle Under Rotation-SimilarityTransformation 69 (5) g + () = R+ R = : Theorem 3.. If to sulement Theorem. by requirement of random BC n+ = (random selection of the oint C n+ ) at condition of uniform robability density distribution, where is random number in the range [ + ; ] with origin in the vortex B n then: (a) distortion between area S F n+ (SF n ) of successor (redecessor) and S C n (S C n+ ) of redecessor (successor) (6) (; ) = achieves minimum = 0 at = at any ; (b) at AC n+ =, the image of rotation-similarity transformation degenerates to circle with 6= 0 at any : (7) AC n = AC (n)ex[g()b] where AC is assumed to be constant, b = =arctan(=); (c) robability of random event (S n < 0 ) exceeds the one (S n 0 ); i:e: (8) at f ( 4=3). Proof. g () g + () (a) Rewrite (5 ) assuming to be random as (9) Sn C = (B nc n ) : Also, write down (0) S F n+ = SF n f = (B nc n ) Insert (9 ; 0 ) to (6 ), so we have () = SF n+ S C n S F n ( + + ) : = + : Function (; ) has minimum = 0 at = (B n C n = B n C n+ ) for any. Plot (; ) for di erent is shown in Fig.. So, invariance relation (0 ) matches (6 ) at = only. At 6=, we should account distortion factor, i.e. () = (; ): (b) If random AC n+ =, jac n j = jac n+ j, so (7 ) obviously describes a circle, where jzj denotes module of number z. In AB n C n, random B n C n+ = = AB n AC n+, i.e. (3) = B n C n+ = + =

5 70 Andrei Moldavanov Solution (3 ) is = 0, which means that can never be, so 6= 0 for any. It also directly follows from the well-nown triangle inequality declaring that any side (B n C n ) of a triangle is greater than the di erence (AB n AC n ) between two other sides.(c) From (4 ; 5 ), comose ratio (4) = g () g + () = Solution (4 ) is (5) f or (6) 4 3 : + + Plots for (; ) on for three di erent and () are in Figure and Figure 3, aroriately. Figure. Deendence measure of distortion (; ) on for three di erent. Figure 3. Deendence of relative robability for folding/unfolding develoment at random transformation conditions.

6 Structural Iinvariance of Right-Angle Triangle Under Rotation-SimilarityTransformation 7 4. Conclusions and outloo We have shown that the family of right-angle triangles with arbitrary legs ratio being under lanar rotation-similarity transformation, demonstrates consistent internal structuring in the successor/redecessor chain at any transformation ste n. In articular, we roved that there is exact equality between the full area of the successor (recursor) triangle and the core area of the built-in core triangle in its recursor (successor), i:e: quantity = is invariant under above transformation irrelevant to the sign of S n. In those cases when 6=, it is ossible to evaluate measure of distortion and adequately describe variations of the structure in the triangles family occurred under above transformation. In this sense, observe the smooth and continuous deendence of on roximity of selection oint to the vertexes B n and C n. Finally, note that robability of the triangles family AB n C n to evolve in two ossible directions, S n > 0 and S n < 0, is not the same. Whereas at the fast mode (f > ), statistically referable direction is the one with S n > 0, at the slow mode (f < ), the develoment with S n < 0 will revail will revail thereby maing referable direction of develoment deending on. 5. Acnowledgements This research did not receive any seci c grant from funding agencies in the ublic, commercial, or not-for-ro t sectors. References [] Bogomolny, A., Golden Ratio in Geometry, htt:// [] Moldavanov, A., Classical Right-Angled Triangles and the Golden Ratio, Forum Geometricorum, 7(07), , htt://forumgeom.fau.edu/ FG07volume7/FG074index.htm. 774 SUNNYBRIDGE DRIVE BURNABY, BC, V5A 3V, CANADA address: trandrei8@gmail.com