Pythagorean Triplet and CVM Quadruplet 1. Pythagorean Triplet over the Empty grid. The Pythagorean theorem connects the sides of rectangular triangle and has Original expression:

h2 = a2 + b2 (1.1) where h — hypotenuse,

a — smaller leg, b — larger leg.

When we draw a rectangular triangle over the Empty grid, their mutual relationship mostly can be determined by the triangle side’s orientation and vertices alignment with CVM points of the Empty grid. The rectangular triangle KLM shown on fig. 1.1 has default position on the Empty grid.

K

h = a=5UN α L b= 16 UN M Fig. 1.1

Now we describe the default position of triangle KLM verbally: a)All three vertices K, L, M are CVM points; b)Small and large legs of triangle have Quad-Orthogonal orientation; c)Large leg LM is triangle’s bottom border; d)The smaller interior angle α ends up with “nose” vertex M, located on right–hand side of the triangle. Shown example of rectangular triangle on fig. 1.1 has two legs measured by integer (whole) numbers of UN’s, but the hypotenuse KM cannot be measured by the integer number of the same units of length. The Quad-Digital Geometry is involved with a special kind of rectangular triangles, which have all three sides measured by integer numbers of the same units of length. If we replace the Original expression (1.1) by numerical values shown in expression (1.2), 52 = 32 + 42 (1.2) then this expression participates in many different occasions in Quad-Digital Geometry and has special name Basic Pythagorean Triplet, see fig. 1.2.a. This is the smallest Pythagorean Triplet. B β 15 10 9 5 6 O 3 α 4 8 C 12 A a) b) c) Fig. 1.2 All three shown Pythagorean Triplets are similar to each other and often called Basic Triplet Family. The Triplet in fig. 1.2.b has twice larger sides than Basic one, and the Triplet in fig. 1.2.c has triplicate the sides of Basic one. Pythagorean theorem expression for Triplets shown on fig. 1.2:

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Fig. 1.2.a: 52 = 32 + 42 ; Fig. 1.2.b: 102 = 62 + 82 ; Fig. 1.2.c: 152 = 92 + 122 . Two other examples of Pythagorean Triplets are presented on fig. 1.3. T M 8 17 5 O 13 O α α L 12 K S 15 R a) b) Fig. 1.3 These two Triplets are “single”, because doubling or tripling of their sizes makes them too large for practical utilization in our cases. Pythagorean theorem expression for Triplets shown on fig. 1.3: Fig. 1.3.a: 132 = 52 + 122 ; Fig. 1.3.b: 172 = 82 + 152 . Pythagorean Triplets shown in fig. 1.3 and larger size Triplets are used in very rear cases. Contrary, triplets of Basic Family, shown in fig. 1.2 are very attractive and valuable is several items of Quad-Digital Geometry. 2. Basic Triplet Family and its Bisector Tandem. In the shown images of Triplets, we already displayed their Bisector Tandems, but we still didn’t bring their characteristics and role in Triplet’s transformations. B 45o B 135o D 135o

O O C A C A F a) b) Fig. 2.1 Incenter O is an intersection point of bisectors of three interior angles A=α B=β C=90o .

The broken line AOB is called Bisector tandem. Angle AOB = 180o — (

135o, fig. 1.2.c and 2.1.a.

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In fig. 2.1.b, we can notice that 180o — 135o = 45o . This is an indication that arms of the bisector are MOP

siblings:

Other Pythagorean triplets, which don’t belong to Basic Family, also have arms of Bisector Tandems that belong to MOP siblings.

In fig. 1.3.a, they are:

In fig. 1.3.b, they are:

3. Transforming Pythagorean Triplet to Equistep one We will work mostly with Basic Triplet Family, see their images on fig. 1.2. They all have similar Bisector Tandems, their arms have the same inclination ratios: for bisector of angle α (smaller acute interior

angle) ρSM =

and for angle β (larger acute interior angle) ρLA =

.

To perform construction of Equistep Triplet, we use two different models, ALPHA and BETA, connected with acute interior angles of the Triplet. E H M B 5 5 α D 3 5 3 4 5 4 β L O α 4 O β 3 C 4 A F K 3 G N a) b) c) d) Fig. 3.1 Triangle ABC, fig. 3.1.a, presents Basic Pythagorean Triplet, having default orientation: its legs have Quad-Orthogonal orientation, its longer leg AC is bottom horizontal border of the triplet and interior angle α has the vertex A which “looks” to the right. Triangle DEF, fig. 3.1.b, is the result of transformation of triangle ABC into Equistep Triplet. Its hypotenuse DE is drawn parallel to the Bisector’s arm AO and, naturally, have the same inclination and CVM step. This transformation belong to ALPHA model. Triangle DEF occupies much larger area than triangle ABC.

Area of triangle ABC equals:

.

One CVM step of DEF triangle is larger than 1 UN by factor ALPHA: = .

Area of triangle DEF equals:

.

Triangle GHK, fig. 3.1.c, has different orientation than triangle ABC. Its shorter side GK is bottom horizontal border of the triplet and interior angle β has the vertex G which “looks” to the right. So, triangle GHK is prepared to be transformed using BETA model. Triangle LMN, fig. 3.1.d, is the result of transformation of triangle GHK into Equistep Triplet. Its hypotenuse LM is drawn parallel to the Bisector’s arm GO and, naturally, have the same inclination and

CVM step. One CVM step of LMN triangle has factor BETA: = .

Area of triangle LMN equals:

.

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4. Orientation of Equistep Triplets and E-star tool Equistep Triplets shown on fig. 3.1. b and d have all sides with Quad-Inclined orientations, and there are some special rules to rotate those Triplets, we are going to investigate them now. We will use two models: ALPHA, see example on fig. 3.1.b, and BETA, see example on fig. 3.1.d. And we will use a visual tool called E-star, which is also divided by two corresponding models. Angle α shown on fig. 3.1.b, is formed with two equistep sides: hypotenuse DE and longer leg DF. Each

CVM step of those sides has length: Just this CVM step is used in each line of the E-star shown on fig. 4.1.a. G C G C α β A F A F default 0 default angle α α angle β β E B E B α β D H D H

a) b)

Fig. 4. 1 As a result, we have four angles and those angles α are used as wrap for angle α of Equistep Triplet shown on fig. 3.1.b. Only question: which of four angles α shown in the E-star is default? Answer: The most left angle α is the default angle in the E-star in fig. 4.1.a. Angle β in triangle LMN shown on fig. 3.1.d, is formed with two equistep sides: hypotenuse LM and

shorter leg LN. Each CVM step of those sides has length: Just this CVM step is used

in each line of the E-star shown on fig. 4.1.b. Similarly, we have four angles and those angles β are used as wrap for angle β of Equistep Triplet shown on fig. 3.1.d. And question again: which of four angles β shown in the E-star is default? Answer: The most left angle β is the default angle in the E-star in fig. 4.1.b. Each E-star, ALPHA or BETA model, shown on fig. 4.1, have four branches: AB, CD, EF, and GH, but those branches play diverse roles in organizing different Triplet’s orientations. Branches AB and CD (red color in fig. 4.1) are called Longest-initial, other two (black color in fig. 4.1) are called Longest-reversed, we will explain and illustrate those terms soon.

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How the E-stars is connected to Triplet’s orientation we are going to see right now. 90o C 90o

E E C A A H H 0o α 180o 0o β 180o

B B G G α β D 270o F D 2700 F a) b) Fig. 4.2 In most cases, we use the Default orientation of any Pythagorean Triplet. This kind of orientation is illustrated twice in fig. 4.2: a) The ALPHA degree of Triplet and E-star see fig. 4.2.a. b) The BETA degree — see fig. 4.2.b. Let’s define the Default orientation of the Triplet. Its Degree identifying angle (ALPHA or BETA) has vertex “looking” to the right, which is symbolized by E-star’s angle 0o. Triplet’s longest side (i.e. hypotenuse) coincides with Longest-initial branch AB of the E-star, so most left α or β angle of E-star wraps the triplet in its Default position. The complete symbol of Default orientation is: INI, 0o . Another Triplet shown on fig. 4.2.a is rotated 90o clockwise, its ALPHA vertex “looks” down, its hypotenuse coincides with Longest-initial branch CD of the E-star, so its orientation is: INI, 90o . Fig. 4.2.b presents two BETA degree Triplets, the first one has Default orientation again: INI, 0o . However, the second Triplet is shown having different orientation. It has the same 90o E-star’s rotation, but in addition, it longest side (hypotenuse) is coincides (reversed) with Longest-reversed branch EF. So, the symbol of its orientation is: REV, 90o . The total number of Equistep Triplet’s orientations is eight: four E-stars angles — 0o, 90o, 180o, 270o, and each of them has two positions of Triplet’s hypotenuse: INI and REV.

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5. From Triplet to Quadruplet Her we repeat the Pythagorean theorem Original expression:

h2 = a2 + b2 (5.1) If we replace the Original expression (5.1) by numerical values shown in expression (5.2), 52 = 32 + 42 (5.2) then this numerical expression represents Pythagorean Triplet and has geometrical image of rectangular triangle with side 3, 4, 5 UN’s or other units of length. Numerical expression (5.2) is mathematically correct if we add one monomial like this: 02 + 52 = 32 + 42 (5.3) But this is not Triplet any more, it has four monomials and can be called Quadruplet. Looking ahead, we can say that there are many real Quadruplets, having four non-zero monomials and all sides are measured with integer numbers. Let’s replace the literal Original expression (5.1) by another literal expression called “Original expression of Quadruplet”: a1

2 + b12 = a2

2 + b22 (5.4)

We put numerical values in order: (5.5) This means that represents the smallest side and — the largest side in some quadrangle. To see practical example of the Quadruplet, let’s replace the Original expression (5.4) by numerical one: 12 + 82 = 42 + 72 (5.6)

Let’s divide the quadruplet’s numerical expression (5.6) by two parts: a) 12 + 82 = H2; (5.7) b) 42 + 72 = H2. (5.8)

Geometrically, numerical expressions (5.7) and (5.8) are represented here by two right triangles, see fig. 5.1. The area of the upper (smaller) triangle is:

SSM =

The area of the lower (larger) triangle is:

SLA =

4UN2

1 8 4 14 UN2

7 Fig.5.1

Hypotenuses of these two triangles are congruent to each other, so these two triangles can be combined by coinciding those two hypotenuses and attached figure will be kind of convex quadrangle. Let’s arrange this attachment using preselected technique showing concrete original orientation, mutual position and relocation of participating triangles. First, we place the larger triangle (having area of 14UN2) into new place having slightly new orientation, see fig. 5.2. Taking into account the role it plays in this occurrence, we call this triangle as Stator — stationary component of this construction.

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1 Stator 8 7 H 7 X Rotor X+Y Y H B R 1 4 8 4 R1

a) b)

Fig.5. 2

The smaller triangle (area 4 UN2) is called Rotor, both Stator and Rotor attached at V-type type point R (see fig. 5.2.a), this point is a coincided vertex of Stator’s larger acute interior angle and Rotor’s vertex of

smaller acute angle, this model is called SL model of construction. The constructed figure is called Quad-joint, the 3 vertices of the Stator remained on CVM positions, only one vertex: vertex of Rotor’s right interior angle landed at non-CVM point. We rotated constructed Quad-joint

counterclockwise by angle

and

constructed Equistep Quad-joint, SL model, see fig. 5.3.

CVM step factor = . Area of Quad-joint ABCD is equal:

SABCD =

(7×4×3.25+1×8×3.25) = 58.5 UN2

Area of original Quad-joint

SOR =

(7×4+1×8) = 18 UN2.

Yes, area of Equistep Quad-joint ABCD is larger than area of the original Quad-joint by 3.25 times. Notice that bisector of obtuse interior angle BCD crosses side AB at non-CVM point.

B 1 C E 8 7 X+Y A 4 D Fig 5.3

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6. SS construction model of Equistep Quad-joint When we constructed earlier the OI Quad-joint, refer to fig. 5.2, we attached the vertices of two interior angles: Stator’s larger interior acute angle X with Rotor’s smaller acute interior angle Y. Now we change orientation of the Stator triangle: both vertices of smaller interior angles Z and Y are attached at point R, see fig. 6.1.a. Stator Rotor 1 8 Z + Y 4 Z Y 4 R B

1 7 8 7 R1

a) b)

Fig. 6.1 In this case: is a total interior acute angle in the constructed Quad-joint, see fig. 6.1.b.

The bisector BR1 of angle (Z + Y) shown on fig. 6.1.b has inclination ratio

.

The OI Quad-joint shown on fig. 6.1.b belongs to SS (Small-Small) model of construction. Let’s compare this Quad-joint with presented earlier SL model of OI Quad-joint, refer to fig. 5.2.b. Technique of their construction is similar (only Stator orientation is different). The angle (Z + Y) in SS model is significantly smaller than corresponding angle in SL model. In general, both SS and SL models look so close in their OI orientation that there is no major different features can be pointed. However, when we move one step farther and transform those OI images into equistep (Quad-inclined all-around) similar Quad-joints (see fig. 6.2), then difference between their SL and SS models will be significant and impressive. B B Cap 1 CVM eye C C α E 8 Nose G 4 α A α A 7 Body F D a) D b) Fig. 6.2 Notice that bisector of obtuse interior angle BCD (see fig. 6.2.a) crosses longest side AB at CVM point, in this case, this point plays important role and we call it CVM eye.

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In fig. 6.2.b, we copied all four outer sides and marked the CVM eye by letter E, then added two perpendicular lines from the point E : EF and EG. Actually, we divided the Quad-joint ABCD by three components, we named them Nose, Cap, and Body. Those three components are Geometric Primitives: a)Nose component has shape of Base Pythagorean Triplet , having sides 3, 4, 5 CVM steps, and smallest interior angle α . b)Cap component has shape of Rectangular Kite with shorter sides of 1 CVM step and longer sides of 3 CVM steps. Angle between two longer sides is α (the same as in the Nose component). c)Body component has shape of Rectangle, sometimes Square, two its sides are equal to smaller leg of the Triplet, and other two — to larger sides of the Kite. Construction of SS model Quad-joint is possible by assembly of proper values of Nose components (triplets) and Cap components (Kites). B 3 9 C KITE E 19 3 9 6 10 F TRIPLET α A RECTANGLE 9 8 6 G 17 9 D

Fig. 6.3 Quad-joint shown on fig. 6.3 is composed by: a) double Triplet (sides 6, 8, 10 CVM steps); b)triple Kites (sides: BC = CF = 3 and BE = EF = 9 CVM steps); c)Body component’s Rectangle EFDG sides: EG = FD = 6 and DG = EF = 9 CVM steps. Numerical expression of CVM Quadruplet corresponding to shown Quad-joint is:

32 + 192 = 92 + 172 (6.1)

The CVM eye point E divides the longest Quad-joint’s side AB by two operating parts: a)Triplet’s hypotenuse AE = 10 CVM steps; b)Kite’s longer side BE = 9 CVM steps.

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Another image of Quad-joint is possible, if instead of Alpha degree direct angle (“looking” to the right), α , we use the Nose component with BETA degree, β . B Cap E C β E Nose α β A M β A K Body K a) b) D Fig. 6.4 7. Cascade of SS model of Equistep Quad-joints Cascade of Quad-joints is a compact Kit of individual Quad-joints with different scale of Cap and Nose components. Those cascades never combine ALPHA and BETA degrees of orientation, only one of them can be used in any cascades of Quad-joints. Here we are going to start visual presentations with samples of ALPHA degree cascades. 17 14 4 11

3 2 8 1 α α 0 Fig. 7.1

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Cascade of Quad-joints shown on fig. 7.1 includes ALPHA degree orientation and size of the Triplet (recall fig. 3.1.b). This cascade has single Triplet’s Nose component and four different Kites’ Cap components. The shortest side of each Quad-joint are marked by numbers 1, 2, 3, 4; and the longest side of each Quad-joint is marked by numbers 8, 11, 14, 17. All those numbers are presented in numerical expressions of corresponding CVM Quadruplets, refer to expressions (7.1) to (7.4). 12 + 82 = 42 + 72 (7.1) 22 + 112 = 52 + 102 (7.2) 32 + 142 = 62 + 132 (7.3)

42 + 172 = 72 + 162 (7.4) Next example of Quad-joints cascade also belongs to ALPHA degree, but this time we have a single Kite’s Cap component and three different Nose components (Triplets). 0 1 8 13 α 18

Fig. 7.2 This cascade has single Cap component (Kite) and three different Triplets in Nose components. The shortest side of all Quad-joints is the same and marked by number 1. The longest side of each Quad-joint is marked by numbers 8, 13, 18. All those numbers are presented in numerical expressions of corresponding CVM Quadruplets, refer to expressions (7.5) to (7.7).

12 + 82 = 42 + 72 (7.5) 12 + 132 = 72 + 112 (7.6) 12 + 182 = 102 + 152 (7.7) Notice that the expression (7.5) includes the smallest Nose component — the Triplet with sides 3, 4, 5 CVM steps.

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The middle expression (7.6) includes larger Nose component — the Triplet with sides 6, 8, 10 CVM steps. And expression (7.7) includes the largest Nose component — the Triplet with sides 9, 12, 15 CVM steps. Now we are going to bring visual presentations of samples of BETA degree cascades and their numerical expressions CVM Quadruplets. 17 6 15 5 13 4 11 3 9 2 7 1 β 0 Fig. 7.3 Cascade of Quad-joints shown on fig. 7.3 is BETA degree cascade. This cascade has single Triplet’s Nose component and six different Kites’ Cap components. The shortest side of each Quad-joint are marked by numbers 1, 2, 3, 4, 5, 6; and the longest side of each Quad-joint is marked by numbers 7, 9, 11, 13, 15, 17. All those numbers are presented in numerical expressions of corresponding CVM Quadruplets, refer to expressions (7.8) to (7.13). 12 + 72 = 52 + 52 (7.8) 22 + 92 = 62 + 72 (7.9) 32 + 112 = 72 + 92 (7.10)

42 + 132 = 82 + 112 (7.11) 52 + 152 = 92 + 132 (7.12) 62 + 172 = 102 + 152 (7.13)

The last example of Quad-joint cascade also belongs to BETA degree, but this time we have a single Kite’s Cap component and four different Nose components (Triplets), see fig. 7.4.

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The shortest side of all Quad-joints is the same and marked by number 1. The longest side of each Quad-joint is marked by numbers 7, 12, 17, 22. All those numbers are presented in numerical expressions of corresponding CVM Quadruplets, refer to expressions (7.14) to (7.17).

0 1 7 12 17 22 β Fig. 7.4 12 + 72 = 52 + 52 (7.14) 12 + 122 = 82 + 92 (7.15) 12 + 172 = 112 + 132 (7.16)

12 + 222 = 142 + 172 (7.17) Quad-joint cascades shown on fig. 7.1 — 7.4 were selected slightly simplified using the following technique: in each case, we used one single component (Nose or Cap) and set of opposite components. Now we are ready to bring more complicated package of components within one cascade. Cascade shown on fig. 7.5 has three different Nose components (Triplets) and three different Cap components (Kites). So, number of individual Quad-joints is nine and we are going to display all nine CVM Quadruplets numerical expressions too. Notice again that we don’t mix here ALPHA and BETA degrees, actually in this example we present only BETA degree SS model Equistep Quad-joints.

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3 2 1 β Fig. 7.5 Now we present the Quadruplet numerical expressions for all nine individual Quad-joints composed in the cascade shown on fig. 7.5.

12 + 72 = 52 + 52 (7.18) 12 + 122 = 82 + 92 (7.19) 12 + 172 = 112 + 132 (7.20)

22 + 92 = 72 + 62 (7.21)

22 + 142 = 102 + 102 (7.22) 22 + 192 = 132 + 142 (7.23)

32 + 112 = 92 + 72 (7.24) 32 + 162 = 122 + 112 (7.25)

32 + 212 = 152 + 152 (7.26) There are still some additional possibilities to construct other Quad-joints and their Cascades. All shown Cascades have Default orientation, but they also can be wrapped by E-stars and have totally eight different orientations similarly to Pythagorean Triplets, recall example on fig. 4.2.

Sincerely, Moses H. Quad.