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Ayuda de DP21

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Los Comandos y Menu, en Help of DP21_MENU


$#+ WPU21 Help Contents

The Contents lists Help topics available for Puertra, Version 21 for Windows 95 and Windows98 (WPU21).  Use the scroll bar to see entries not currently visible in the Help window. To learn how to use Help, press F1 or choose Using Help from the Help menu.


Main Topics


Getting StartedGetting_Started

Formal RulesFormal_Rules



Rige EditionRige_Edition

Rige FinishingRige_Finishing



$#+Getting Started

Using Puertra is easy: just by running it, a Rige is loaded (INI.RIG) and a TE appears. You can move it across the surface, change its size and go through all Menu items, seeing the result on the display, and so learning the use and meaning of the CommandsCommands. The Formal RulesFormal_Rules are a set of descriptive properties of Rige that summarize a Tradition, a way of designing Rige as well as its Aesthetics.

            The modification of an existing Form, that is the LatticeLattice and RigeRige, is somewhat more difficult since you must first have understood the meanings and relationships between AnglesAngles and DistancesDistances, as well as the SymmetriesSymmetries applied to the Generative ElementGenerative_Element and the Translation ElementTranslation_Element.

            The last and final step is the design of new Riges inside new Lattices, with or without MetabolaiMetabolai; you need some knowledge of elementary geometry and, above all, good taste and artistic feeling: Art is based on a sensible choice among thousands of possibilities, and only the mentioned qualities can guide you to reach beautiful forms, as the former inventors of Rige did, with infinitely less material resources (such as Puertra!) than us, but probably with infinitely more coherent and meaningful ways of living.


PUERTRA is a package developed by Aldebaran Soft, that generates, draws, edits and colours the Islamic Geometric Interlaced Lattices, Rige, which we label Doors of Tradition, because these Rige are often found on mosque doors and windows. These forms express, in numeric symbolism, relationships with the Abstract and the Absolute.

The nucleus of PUERTRA is a theoretical model of the Rige, consisting on two main parts:

          1. The LatticeLattice, consisting of a series of parallel straight lines, regularly spaced, and intersecting each other in nodes, forming segments between every two nodes.

          2. The Generative ElementGenerative_Element  (GE):  is a selection of some of these segments in order to form a pattern inside a triangle. By rotating this triangle round a centre, a form with circular symmetry is created, the  TE (Translation ElementTranslation_Element), which, repeated by translation in two dimensions, covers the plane.

The design is made in five stages:

          1. Didactic Sketch of GE.

          2. Circular Repetition of GE, becoming the TE.

3. Plane covering by translation, with colours and interlacing.

          4. Colored background.            

          5. Framing of the form with a variety of patterns fitting in with lattice properties.



          The forms and designs described above can be determined and actually drawn by the program PUERTRA, in an easy and didactic way. PUERTRA, acronym of PUERTAS de la TRADICION (DOORS of TRADITION), allows the user to choose order, distances, angles, segments, interlacing, frames, colours, and any other formal aspect of a Rige. This Help facility  lists the main commands of PUERTRA. Several Submenus inform on other possibilities as well.

          With this assistance, beautiful and didactic forms can be designed, and printed with any of the available programs to capture, display, graphics-format convert, edit and print in black/white or color. All the figures that appear in this Help have been designed with PUERTRA, unless said otherwise.

          During the program running, sizes, proportions, colours and textures, for both background and frames, can be changed on a trial and error basis. A more careful edition can also be chosen, where the main parameters of the form can be changed to modify or create a new one. Any Rige can be stored on disk for later recovering with all its original parameters; with this facility, a big library can be created ‒each Rige occupies only 1-4 Kbytes.

                                                                                                 Aldebaran Software. 1999



The lattice is a family of families of straight lines. Several straight lines compose each family, all parallel among themselves. The DistancesDistances between each pair of neighbouring straight lines take only a few different values, usually two. These values are linked to the lattice OrderORDER as well as the AnglesAngles between families.  In simple cases, there is only one sublattice in a Lattice, and, the same series of distances for all the Angles. But in complex forms, it is possible to find 2 or 3 Sublattices; in each one, several Subseries of distances according to its different angle, and in each one of this subseries, different number and values of these distances. See Lattice EditionLattice_Edition.




The order of a lattice, N, is a natural integer. This number is the number of different directions present in the lattice, or, equivalently, the number of AnglesAngles. The angle between any two directions is a multiple of a single angle, the Basic Angle, which means that these directions are parallel to the radius of a circumference divided in N parts. The order is usually even. The most usual values are 6, 8, 10 and 12. Others also found are 20, 24, 32, 36. But 14, 28, and big numbers are extremely rare.

          In theory, the order of a lattice could be any integer, but constructive, psychological, aesthetic and mathematical reasons limit its values to small numbers with many divisors. Typical cases are 8, 10, 16, 20, 24 and 32. The constructive reasons are (or were) the difficulty of making wood, plaster, ceramic pieces with extremely fine angles, and with too many possibilities: in a 36-lattice there may appear angles of 10, 20, 30, ... 350 degrees, 35 different values. The tools would have to have been very exact, which was not easy in medieval times. The psychological reasons relay on the difficulty in perceiving so many different angles and distances. This leads us to aesthetics: the sense of unity is lost as in an intricate and baroque relief. Some mathematical reasons can also be found, probably related to the former: the higher the order, the more polygons are possible and the lattice will need many different distances, complicating again the form for designer, maker and observer.

          Continue studying the  DistancesDistances in relation to the Angles for a specific Order.


As we said in OrderOrder and in LatticeLattice , all the angles that appear in a Rige are multiples of a basic one, which value is the Nth part of the full circle, that is, 360º/N, being N the Order of the Lattice. All the segments in the Rige are parallel to one of these angles, except when the Rige includes more then one Lattice, thus becoming what we have called Metabolê ‒plural: MetabolaiMetabolai. Continue studying the  DistancesDistances in relation to the Angles for a specific Order.



Let us expand further this subject of distances between the straight lines of a family. If we do not impose any symmetrical rule, any distance is possible. But if we do, as Tradition demands, once we choose one, the gyration for central symmetry will create 'alias' in another radius, thus creating new distances as a necessity. Let us see some cases from the simplest to some more complicated ones.

  ORDER 6. This is a very simple case. The angles are the 6 multiples of alfa=360°/6=60° that is, 0º, 60º, 120º, 180º, 240º and 300º. If we consider a triangle, it must be equilateral and the angles will be 60º, 60º, 60º. If we choose any distance on the horizontal as a side, the same distance appears by rotating it on the oblique side, for equilateral. Thus we have only one distance necessary in the case N=6. We can use more, of course, but the order does not impose it, one is enough. See Fig.1.   

   ORDER 8.  It is a simple, but already interesting case. The angles are the 7 multiples of alpha = 360°/8=45°, that is, 45º, 90º, 135º, 180º, 225º, 270º and 315º. If we consider a triangle with angles 45º-45º-90º, once we gyrate the hypotenuse on the horizontal side, we obtain a new distance as a consequence of the first one. The ratio between them is V2/1 =V2 (V means 'square root of'). So the lattice order implies a specific distance ratio. See this ratio in Fig.2. Now the problem is to know how many different distances we need. If we raise another perpendicular to the horizontal at point V2, the new distance on the diagonal will be V2 x V2 = 2. But this distance is 2 times 1, that is, a distance equal to the first, 1: we do not need another distance in this case. In general, any gyration of a linear combination of these two distances, 1 and V2, will provide us with another combination of them: they form what is known as a Vector Space, with 1 and V2 as a Base of the Space:  (a+bV2) x V2 = 2b + aV2 = b' + aV2.        

 ORDER 10. A very interesting and very beautiful case. The angles are the 10 multiples of alfa=360°/10=36°, that is, 0, 36, 72, 108, 144, 180º, and their axial symmetries with the  horizontal. A simple triangle will be the isosceles 36º-72º-72º. Once the base of the triangle is chosen, the distances between parallels that appear now in the triangle are its heights. The ratio of their values is found to be, after a little counting, 1.6180.... But this is a prestigious number, older than the civilizations about which we are talking: this is the "Golden Proportion", considered as a very harmonic ratio of two distances (like a picture frame, for instance). Its exact value is (1+V5)/2; let us call it PHI. If now, we take another triangle with the smaller height equal to a linear combination of 1 and PHI, with coefficients a, b, the other one will be:  (a+b PHI) x PHI = a PHI + b PHI x PHI.      But       PHI x PHI = (1+V5)/2 x (1+V5)/2 =  (1+5+2V5)/4 = (3+V5)/2 =1+PHI    therefore  (a+ b PHI) x PHI = a PHI + b (PHI  x  PHI) = a PHI + b (1+PHI) = b + (a+b) PHI = b + a' PHI    what means that also the other height is a linear combination of 1 and PHI, and, again in this 10-space, the pair (1, PHI) is a base vector. For instance, in Fig.3, all the distances are equal to 1, PHI, 1-PHI or any other linear combination. Beautiful and well-known relations can be found for this number PHI; see several developments of this subject in [Coxeter,1973. Ghica, 1977. Rademacher & Toeplitz,1970].

         GENERAL CASE. As we have seen in the former cases, any order which engenders in other radii distances which can be expressed as a linear combination of two, will accept this pair as a base, and any linear combination of them, or what is equal, the distance between any two straight lines of the same series, will form a legal or permitted triangle (with vertices on some nodes of the lattice), by using  some other combinations of the base vectors.

          Since any elemental figure (EE) of a Rige can be decomposed into triangles, the found basis also allows legal Rige, and therefore constitutes answers (not all, but some) to the question raised about the distances between straight series. In Table 1 we can see some of these answers. In it are also shown several possible polygons in each lattice, some of which were shown in the contiguous figures. In Table 1 are listed base vector ratios for greater values of N, obtained by the former expression, which are also equal to the value 2 cos(p/n), deduced by simple geometric calculus.






a/b exact

   6  1 1
   8  1.4142.... V2 / 2
  10  1.6180 ... (V5+1)/2
  12  1.7320... V3
  14  1.8019...  
  16  1.8477...  
  18  1.8793...  
  20  1.9021  






Symmetries are an essential characteristic of Rige artistic design. They create internal similarities within the form itself, which so acquire a psychological pregnancy, that is, attract the eye interest and becomes centers of attention, able to convey meanings, to become symbols. From the geometric point of view, three types of symmetries appear in Rige figures:

1.  Circular: with a point that acts as a center around which all the figure is equal to itself when rotated a determined angle. This angle is always an integer division of the full turn (360º); lets call alpha this angle, which value is alpha=360º/N, N being the OrderOrder of the Lattice or the order of the first lattice, when you have several for the same Rige. The figure takes an aspect of flower or sun. Of course, all the multiples of alpha are also symmetry angles: the figure admits many kinds of turns, so becoming a rotating figure for the eye.

2. Axial, with a straight line that acts as an axis, around which the figure is equal to itself when rotated 180º around the axis. This turn occurs in the space surrounding the figure plane; the figure must came out of this plane to arrive back onto it again after the half turn, as a hand must do to coincide with the other one. But, also as with the hands, the turned one presents the reverse surface to the eye (the turned with finger nails, the other without). So the covered FramesFrames become the covering, and conversely.

3. Central: two symmetric points are on a straight line divided equally by the Symmetry Center.

            The three types appear frequently in any Rige. 

There is an aesthetic (unexpressed) rule: all the Elementary Elements (shapes between frames) should admit at least an axial symmetry. It happens very often, indeed, though it is not always obvious.


$#+Generative Element

It is a selection of some of these segments in order to form a pattern inside a triangle. By gyrating this triangle around a center, a form with circular symmetry is created, which, repeated by translation in two dimensions, covers the plane. Each Rige is codified by means of:  1. The LatticeLattice parameters, that are the distances between parallels, and the number of directions in a regular circle division. See Lattice EditionLattice_Edition. 2. The Rige parameters, the polygonal lines that form it. A polygonal is defined by the succession of its straight lines, defined by the ordinals on the Lattice: order of the direction from 0 (horizontal) to N/2, and order of the distance to the center, from 0 (passing on center) to any number. These ordinals appear when the Rige is drawn in a preliminary sketch.  See Rige EditionRige_Edition


$#+Translation Element

By turning the GE (Generative ElementGenerative_Element), a new pattern is obtained: the TE (Translation Element). This is a form that will be copied by translation in two directions, horizontal and vertical; so, they cover the plane when its shape is square or rectangular, as bricks making a wall.  The separation between the centers of two contiguous TE is usually equal to the actual corresponding dimension of the TE, which gives a factor of 1. This factor can be augmented or diminished, changing accordingly the respective separations between TE.




A metabolê (plural metabolai) is a Rige where there are two o more LatticeLattice, usually dominating in different zones. Interesting phenomena appear in the boundaries between them. Thus, by Metabolê, a term taken from the old Greek theory of music, we understand a local change in the lattice of a Rige. That directly violates our R2 rule; however the rule holds, but in an specific area, while another lattice becomes valid in the new one. In fact, a unique lattice can comprehend both local lattices, and no concept of metabolê should be required: only the specific use of this universal lattice by a Rige would then focus on some straight lines or others. But there are some justifications for that new concept: first, perception selects easily two areas in an actual metabolê; second, the combined lattice would became very complex, and underused.

        Several types of metabolai can be considered: the first, the simplest one, consists of only a turn of the first lattice; it is the same, but with a different reference or direction 0. An example can be two concentric 8-lattices with a circular shift of half-basic angle. See in Fig.3 this metabolê and its beautiful effect, in a Rige that can be found in the Alhambra, Granada (Spain). The lattice could be a unique 16-one, but also two concentric 8's. The choice would depend on the simplicity of computer implementation (see below).

          A second type of metabolê, also simple, can appear: a change, which keeps reference angles but jumps to a multiple or sub multiple order. In those cases, both lattices share the main directions, but one of them introduces another set of main directions, in a symmetrical way, as the new directions will be inserted between the old ones. For example we can change from 8 to 16, or from 24 to 12. As before, a global or universal lattice can include both if desired.

          A somewhat more complex metabolê, similar to the former, is a change to an order which share some divisor, their integers have a maximum common divisor, as in the cases 8 to 12 (m.c.d.=4) or 9 to 24 (m.c.d.=3). In this case, a global lattice can include both of them, too.

          The most complex metabolê consists of a change between two prime orders, like 12-13 (never found by the author!). However, usually more simple changes appear, as between numbers which share only the factor '2' , as 10-12, in a wood door in the National Museum in Baghdad. Another rare example can be found in the Alcazaba museum of Malaga, where a Nazari relief (about XV century) shows a rare metabolê from a central part on 8-lattice, and 4 parts on a 7-lattice. This example is, perhaps, a "tour de force" from an Art in decline. But an even more complex example found by the author is the one in Konya, Turkey, which uses three different lattices, 16, 12 and 10, that coexist harmoniously here.

          The metabolê has an exemplary value for us: it changes references, and/or circle divisions, in the same way that musical metabolê change the reference, in tonal or modal tonic, or the division of octave or tetra chord: genus Rast on note rast can change to Rast on naua, reference change, or to Nahauand on note rast, division change [Barker,1989. Sánchez, 1993-a]), The Rige metabolê, like its musical counterpart, allows keeping simplicity while introducing variations: the psycho-aesthetic effect is to allow easy recognition and appreciation, which becomes denser after some study, as with some masterpieces.


$#+Formal Rules

Formal Rules of RigeRige


  We formally characterize the Rectilinear Lace by means of the following restrictions or rules for the Rige forms:

          R0. A Rige is developed in two dimensions, however, relieves and holes are used to mark the designs.

          R1. Only figures limited by straight segments are considered. Curvilinear design can be used to fill elementary polygonal figures, to avoid rigidity, dryness and too geometrical forms. But they do not contradict the basic rectilinear criterion, which remains.

          R2. All these segments are parallel to a few directions in the plane, derived from the division of the compass (360º degrees) by an integer N. This integer is called the OrderORDER , and the N directions, the main directions. Direction 0 will be the Reference, and  360º/N will be called the Basic Angle. All the AnglesAngles in the Rige are multiple of the basic Angle. Violations of this rule are exceptional, and will be considered in the MetabolaiMetabolai , section below. They not, however, affect this very basic law.

          R3. All these segments are situated on specific straight lines, parallel to the main directions. The DistancesDistances between those lines are always equal to a very small number of quantities, 1, 2 or 3, according to the order. These distances are not arbitrary, but are determined by this order and by the angles - all multiples of   - between the main directions.

          R4. The set of the N families of parallel straight lines is called the LatticeLATTICE or Net of the form, briefly the N-lattice. All the segments belong to the Net; all the figures belong to the Net; the Net can be considered as the union or set of many possible Rige forms of a given order.

          R5. The segments form closed polygonal figures, convex or not, symmetrical or not. Those without segments between them will be the Elementary Elements -EE- (elements in the lowest level), which, by juxtaposition, give larger figures. See DistancesDistances.

          R6. Their sizes are not too different: a general aspect of constant density of lines can be observed (and measured), recalling a grid or canvass.

          R7. At least one point can be found which is a symmetry center for all, or a part, of the form. The order of this central symmetry is equal to the order of the net to which the form belongs.

          R8. The segments that limits the figures are united in polygonal lines (Polygonalspolygonals) with a width, as FramesFrames or sheets, which are never interrupted: whether they are closed or whether they finish in the external limits of the figure (actual figures are always limited, while nets are infinite by nature).

          R9. The width of the frame is constant for the same form.

          R10. When frames intersect, only two do so at the same point. The angle is nonzero (and of course, is equal to the angle between two main directions). Frames never bend at the intersection point, or, equivalently, opposite angles are equal.

          R11. The intersection is usually represented (from Arabic countries to the West) as in three dimensions: one frame covers the other.

          R12. When one frame (f1) covers another (f2) at a given intersection, at the following intersection it will be covered in turn by a third (f3) (which could be the second, f2).

          R13. The form can actually be constructed by means of physical frames, bent and interlaced according to the former rules. Simple figures (EE) appear as empty spaces between frames.

          R14. When the widths of frames are maximum, they fill the entire plane, with the disappearance of elementary figures. The aspect is now a polygonal chessboard, and it is used in woodwork, musical instrument decoration, etc.



See it in Help of DP21_MENU

$#+Spatial Surface

The Rige is usually drawn on a plane; but it can be also done on a different surface. Puertra does it on a sphere, in which case four alternative projection methods of a plane on the sphere can be adopted. These methods are titled s1, s2, s3, s4. The user should try them out to choose the one he considers adequate. (s3 is taken by default). Commands:

SECTION    Action                            Increase          Switch/Rotate           Decrease         Introduce

     Spherical/Plane        Dome                                                                                                                                 #

     Radius of Spher.Dome                               ALT+a                                                ALT+s

     Steps Num.Spac.Straight                            F9                                                      F10


The Rige can be conceived as a set of intersected polygonal (PolygonalsPolygonals) frames or sheets. Frames are infinite, never ends inside the figure: either they close on themselves, as regular or irregular rings, or they are cat by the limits of the figure, thus suggesting a further propagation. Ar we see in see in Formal RulesFormal_Rules, they can seem to intersect covering one to another in an alternate basis (if one covers another in covered in its next intersection. The SymmetriesSymmetries and Translations preserve this rule.

            The interlace is a nonessential feature for the Rige geometry, and in the eastern countries (from Persia onwards) does not appear. However it is common in the Arabic countries and was in the Spanish Al-Andalus and afterwards, in Christian commissioned Mudejar art (XIII- XVI cent.).

          The universal way to achieve this interlace consists in passing the frame alternatively up and down the frames which it crosses. This system always has a solution; it is always possible to interlace frames in an alternative up-down basis.

Indeed, if we consider the polygons limited by the frames, we see that their sides are always segments of frames between two crosses, or, if the frame bends, the fragment between crosses will give more sides, as shown in the schema. Let us consider as positive the polygon whose sides begin in an up position and finish down when going in the clockwise direction of gyration , and negative the opposite, as can be seen in the schema. Since only two frames intersect at a time (R10), their crossing creates four angles, two contiguous polygons sharing only one segment of frame, and four polygons sharing a node. Now, it is easy to see that, if we select a positive polygon, those contiguous to it will be negative, and the other one, opposed by the angle, positive. That gives give us a rule to intersecting: we choose one polygon as positive and all the others will take their respective sign; the frames will be then accordingly interwoven.



ARDALAN, N. & BAKHTIAR,L.The sense of Unity. Chicago Press. Chicago, London, 1973,1979.

AUDSLEY, W & G. Designs and Patterns from Historic Ornament. Dover. N.York.1968.

ABOU-ESH, I.M. Design Concepts of Islamic Arquitecture. Dar Al-Arabiyah. Beirut, 1970.

BARKER, A.(ed):Greek Musical Writings,book II:Harmony&Acoustic Theory. Univ. Pres. Cambridge. 1989

BOURGOIN, J. Arabic Geometrical Pattern & Design. Dover, N.York. 1973.

BOURGOIN, J. Islamic Patterns. Dover, N.York. 1977.

BURCKHARDT, T. Moorish Culture in Spain. Allen & Unwi. Lond,1972.

BURCKHARDT, T. Art of Islam. Language and Meaning. World of Islam Festival. London, 1976.

CABANELAS, D. "El Techo del Salón de Comares en la Alhambra". Patronato Alhambra. Granada, 1988.

COXETER, H.S.M. Regular Polytopes. Dover, N.York.1973.

CRITCHLOW, K. Islamic Patterns. Thames & Hudson, London, 1992.

GHICA, M.C. Estética de las Proporciones en la Naturaleza  y en las Artes. Poseidón. Barcelona, 1977.

GRABAR, O. Los fundamentos del arte islámico. Cátedra, Madrid, 1984.

JAWAD al-JANAB, T. Studies in Medieval Architecture. Ministry of Culture & Information Baghdad, 1982.

MARTÍNEZ, B. "Carpintería Mudéjar Toledana". Cuadernos Alhambra,12. 1976.

NUERE, E. La Carpintería de Armar Española. Ministerio.Cultura. Madrid, 1989.

NASR,S.H. Islamic Art and Speirituality. Golgonooza. Ipswich, 1987.

PHIGEL,S. Anadolu Selçuklulari'nin Ta  Tezyinati. Türk Tarih Korumu Basimevi, Ankara,1987.

PAVÓN MALDONADO, B. El Arte Hispano-Mulsulmán en su Decoración Geométrica. Min. Cultura y  Agenc. Española. Cooperación  Internacional. Madrid, 1981, 1990.

PRIETO y VIVES, A. El Arte de la Lacería. Coleg.Ingen. Caminos Canales Puertos. Madrid, 1977.

RADEMACHER & TOEPLITZ. Números y Figuras. Alianza, Madrid,1970.

SÁNCHEZ, A. Trigonometría Rectilínea y Esférica. Lib.Romo. Madrid, 1944.

SÁNCHEZ, F.J. "ESCALA, Automatic Measurement of Oriental Scales". Proc. ICEMCO. Univ. Cambridge. London, 1993a.

SÁNCHEZ, F.J. (1993-b.) "PUERTRA: a Model of Islamic Rectilinear Interlaced Lattices". Proc. 4th. ICEMCO. Cambridge: Univ. Cambridge. 

SÁNCHEZ, F.J. (1995)  "A Model of Islamic Rectilinear Interlaced Lattices". Interactive Internet version:  <>

SÁNCHEZ, F.J. (1996.) "Lacework Analysis: Three Tilework Panels from Isfahan". Proc. 5th ICEMCO, Cambridge. Cambridge: Univ. Cambridge. 

SPELTZ, A. The Styles of Ornament. Dover, N.York. 1959.

VARDERBROECK, A. Philosophical geometry. Inner Traditions Inter.  Rochester, 1972.

WILSON, E. Islamic Designs. Dover. N.York. 1988.


$ Contents

# Contents

+ 00001

$ Getting Started

# Getting_Started

+ 00002

$ Rige

# Rige

+ 00003

$ Lattice

# Lattice

+ A:00001

$ Order

# Order

+ B:00001

$ Angles

# Angles

+ B:00002

$ Distances

# Distances

+ B:00003

$ Symmetries

# Symmetries

+ B:00004

$ Generative Element

# Generative_Element

+ A:00002

$ Translation Element

# Translation_Element

+ A:00003

$ Metabolai

# Metabolai

$ Formal Rules

# Formal_Rules

$ Commands

# Commands

+ 00004

$ Plane Transformations

# Plane_Transformations

+ C:00001

$ Block Displacements

# Block_Displacements

+ C:00002

$ Spatial Transformations

# Spatial_Transformations

$ Spatial Surface

# Spatial_Surface

$ Frames

# Frames

$ References

# References

+ 00006


$#+Rige Edition

The first step to edit (change an already existing Rige is to change the Lattice on which the Rige is drawn: this is done by its Lattice EditionLattice_Edition. The edition of a RigeRige itself consists in the modification of some of the  PolygonalPolygonals lines that compose it, and in the choice of the covering polygonal in each intersections, that is the Polygonal Drawing Order. The Edition utility directs you through all these polygonals and prompts you to accept, for all their segments, their AnglesAngles and DistancesDistances index ‒their values were given in Lattice Edition. You can change the proposed index by using the '+' and '-' keys, which vary accordingly their values; or you can type the desired values. It is also possible to 'kill' some segments or to 'introduce' new ones, as well as to 'part' some polygonals in two branches, copy them for independent edition,

          You can also choose the order in which the polygonals are drawn; this implies which polygonals cover the others at their intersections: any one is then covered by all the following. This choice is not always easy, and sometimes you must cut one of the polygonals in two halves: think in three intersecting straight lines, as they appear in the figure included in the FramesFrames topic (the two rectangles in the right side): one of them must be halved in order to comply with the "one above, one below" rule, the R12 in  Formal RulesFormal_Rules. This also happens in the opening Rige INI.RIG. Edit its eight polygonals and see how this frame halving achieves a correct covering.

  This reordering has been made easy by Puertra ‒ with respect to previous versions‒, by numbering each polygonal on the screen, so allowing the user to alter the ordering in a friendly interactive basis.

   So, in its maximum complication, a figure designed with Puertra can include several Rige; each Rige can include several Sublattices with their own parameters. On these Sublattices, several Polygonals are chosen, each one composed by several Segments.

$#+Lattice Edition

            Lattice Edition is the definition of the LatticeLattice on which the RigeRige is placed as on a frame, choosing some of its segments to draw the sheets or frames which compose this  Rige. The Lattice is determined by the repetition by symmetries of the GE ( Generative ElementGenerative_Element ) to form the TE (Translation_ElementTranslation_Element) and the repetition by translations of this TE to cover the plane (or some other surfaces). Therefore, the lattice edition is in fact the GE edition, which consists of the choice of the angles (Order), distances, size, etc..

            The GE can include only one lattice, or several, in which case, the first of them is the one which defines the limits and symmetries of the GE, the others having their centers, situations and symmetry properties, inside the GE boundaries.

            By typing '0' or clicking the 'Edition-Calcu' menu you access the standard Windows calculator, in which you can calculate appropriate values for your DistancesDistances.

          All the required parameters, some particular of the GE some particular of the TE some of the lattice itself, are explained on the Input Box itself. Distances are changed gradually on the screen to adjust their mutual situations when the input values do not fit as expected, or to rest new combinations. Thus the lattice can be designed by exact values or on a trial and error basis.

$#+Gyrated Translation

Usually the Translation ElementTranslation_Element is displaced parallel to itself. But sometimes you must turn it with each translation to achieve different figures. This is done for instance in a beautiful 'leaves' design found in Alhambra of Granada (Spain), where each translation is accompanied by a half turn ‒both for black and white leaves. You can also change it during Lattice EditionLattice_Edition.


$#+Circular Repetitions

Usually the  Generative ElementGenerative_Element is rotated spatially , that is, turned around an axe, as book leaves, forming axial SymmetriesSymmetries. But this turn can also be on a plane, becoming only a circular symmetry, the figure thus taking a swastika aspect, as the one besides. This effect is also achieved with key '!'.



Any Rige is composed by several interlaced polygonals. A polygonal is composed by several straight segments connected at their extremes (if they compose a closed figure, it is called a polygon). These segments belong to the lattice straight lines.

Each lattice line is defined and coded with two parameters: 1. Its Angle with the reference or basic angle (usually 0º but sometimes another one); and 2. Its Distance to the Center, an arbitrary point (usually the figure center but often another point).           Therefore a segment is defined by means of 3 straight lines, the one to which the segment belongs (the line) and those that intersect it in the segment extremes (ends). The segment coding consists of these three straight line codes in the order: end, line, end. as can be seen in the first figure.

          The polygonal is then easily coded by listing its segment codes: and since any second end is the line of the next segment, only the first end of the first segment and the second end of the last segment are needed besides the lines of each segment. For instance, in the second figure, the lower horizontal segment, the 'green' one, is coded as follows: (1,0) (angle 1, in 8-lattice, so 45º; and distance 0, crossing the center. (0,-1) (angle 0, 0º , the reference angle; and distance -1, since it is the first after the one passing on the center ( the negative sign means that when you go away from the center along this line, the center is on your right; it would be positive if it fell on your left). The following line, the second end of the segment, is the green vertical: (2,3), since its angle is the second in 8-lattice, 90º, and its distance the third, positive because the center is on its left.

          But you will see that with this coding, the union of both frames would have an angle of 45º, instead of the 135º, to link correctly with the vertical line. The reason is that you change from negative distance to positive: in that case you need to insert an extra code that 'bends' the frame: the code is the pair (-1,-1) meaningless as line code. Thus the complete sequence is:  ( 1,0) ; (0,-1) ; (-1,-1) ; (2,3).  (parentheses, commas and semicolons are used here for clarifications, they are not actual codes).

          There is an additional extra code, the pair (-3, -3); it happens only when there are more than one lattice in the figure and you change from one to another, so that angle and distance codes mean different lines. The code change the lattice to the next, coming back to the first when you reach the last one). You can also use the index of the desired lattice, i.e., 'k', as the second number of the pair, which becomes then (-3, k).

          However, in Lattice EditionLattice_Edition you are helped by Puertra to make your choice since the selected line blinks, so you can see if it is the desired one, and change it if not.

          Finally, remember that, in Puertra, every counting begins at 0, not 1 ‒the first distance is the distance 0, the second the distance 1, and so on. An exception is the polygonal order of drawing, which begins in 1. Sorry!


Any RigeRige can be stored in and retrieved from a disk as files with the extension RIG, such the initial one, called INI.RIG, loaded at start time from the directory where Wpu21.EXE is placed ‒otherwise a warning is issued, and the user must find and open by himself another RIG file to run the program. The file INI.RIG is provided within the PUERTRA package, but the user can choose another and call it INI.RIG.

            At any time during the running of Wpu21, the user can store the actual Rige, with all its parameters, into a new file with a chosen name. He can make subdirectories to classify his/her RIG files according to their properties (i.e., according their Order or their Color).

            The RIG file is written in ASCII code and can be seen in any ASCII editor (such as NOTEPAD in Windows) and even modified there; however only experienced users should do that, the proper way being to modify sizes, colors, parameters with the user CommandsCommands , Lattice EditionLattice_Edition and  Rige EditionLattice_Edition.

            For stored Riges, the user must be aware of the printing properties of numbers in the the Operating System: specially the decimal separator in fractional numbers must be a full stop or point ( . ), rather than a comma ( , ), otherwise an error would occur. Thus 2.23 is acceptable, but 2,23 is not.

            The Graphics patterns generated by Puertra can be grabbed by the 'PrintScreen' Windows utility, pasted in any Graphic Editor (such as PAINTBRUSH or PSP) and be modified there. Graphic files (such those in BMP, TIFF, GIF or JPG  formats) can then be stored, printed or included in the user's documents (as this Help file shows).

$#+Rige Analysis

          Now you know the concepts used by Puertra in RigeRige study. But, how to interpret them in the analysis of an actual Rige found in a Mosque wall, for instance?.

          The analysis will follow the following steps, divided in sections: LATTICE, Rige, COMPLEMENTS.


1.  OrderOrder of the LatticeLattice on which the Rige is chosen. Count how many different AnglesAngles are present in the analyzed form, and find the minimal circle division that produce these angles with respect a reference angle (0º or otherwise). The found divider is the Order we were looking for.
2. Find the DistancesDistances  between the family of parallels to each one these angles. Check if all this distances can be expressed as linear combinations of only 1 (rare), 2 (probable, look for it), or more (probably unnecessary). Establish the series of these distances for each one of the angles, and check if the series are equal, or equal to 2 types (alternate), or more. These are the Subseries you are asked for during  Lattice EditionLattice_Edition.
 3. Find the Center of your figure. Probably it is the obvious center of a highly symmetrical form; but sometimes you must choose it among several indifferent possibilities. The center is important because it is the Rotation Center of several SymmetriesSymmetries that usually appear in the Rige subject.
 4. Find the GE (Generative ElementGenerative_Element ) of your form. This is the minimal part ( or one of the minimal parts) that generates the symmetrical form or one of its parts. The rotation of this GE around the center, either axially (see  SymmetriesSymmetries ) or by means of Circular RepetitionsCircular_Repetitions generates the mentioned symmetrical form, which must now copied be translation, thus becoming the TE:
5.  Translation ElementTranslation_Element (TE) It can be simply translated to cover the plane, or, sometimes be submitted to a Gyrated TranslationGyrated_Translation . The horizontal and vertical distances between contiguous TE's must be ascertained, as well as the occurrence of a displacement between rows of TE.
6, Once you have defined the GE and the TE, you have the Lattice (with one or several Sublattices).   
II. Rige
7. On the canvass or net that crosses the GE, you must choose several PolygonalsPolygonals  which, intersecting and interlacing reproduce the actual case being analyzed. You can use long Polygonals composed by many segments (up to 25) or very short ones, composed by only 1 segment ‒codified however by 3 segment codes: End, Line, End, as it is explained in the Polygonals topic.
8. Now you must determine the Draw Order (please do not confuse this order with the Lattice Order seen in Step 1 !)  in which you place your polygonals. This is easy with short ones, but even then you must have a clear numbering of this Draw Order to avoid repeating the process in Rige EditionRige_Edition. If you define your Polygonal in the correct draw order, it is not necessary specify another one when prompted for it after the Polygonal coding.
9. You can type 'r' on the Puertra display to see the TE develop until the entire plane is covered.



10. Once you have defined the essential characteristics of the Rige, some complements must be chosen to complete the global visual effect. They are described in Rige FinishingRige_Finishing, BackgroundBackgroundColorsColorsBoundariesBoundaries Manual DrawManual_DrawLinesLines, and  BoundariesBoundaries.


$#+Rige Finishing

The Rige and its Lattice are already defined. But there are still several operations to be done before it takes the desired aspect. You can change all the  ColorsColors that compose the FramesFrames, and the BackgroundBackground.

          You can also limit the figure by predefined BoundariesBoundaries, necessary to shape properly the Rige. You can also make some small touches by  Manual DrawManual_Draw.



It is made with a solid color covered by small lines ‒like a texture-. You can change the ColorsColors of both. But you may use another Rige as a textured background: take one, even the same and make the scale much smaller than the main Rige. If the scales of both are in a simple relationship (4, 8, 16) the same figures of the background Rige will appear in the same elements of the foreground Rige. Occasionally you can use another ‒even the same‒ Rige as a texture background, that is, a small size figure that contrasts, in color, form, order with the upper form, as the figure shows. Results are more harmonious when simple ratio sizes are used; but non-related sizes could fill the texture function better. Try.




Color is a art in itself. A judicious use of it is necessary to avoid a too colorful and childish combination. The problem is how to reduce the enormous computer possibilities to reach an harmonious combination. Let see some suggestions:

1. Use as few colors as possible. Two are the minimum, to have figures against a background.

2. Contrast frames against the rest.

3. Contrast not only colors themselves (hue) but also their luminosity. This is achieved when the figure stands even when converted to black and white.

                                                                   According to this philosophy, only six simultaneous colors are used in our Puertra program. They are called Item Colors, and their names are: 1. Lines, 2. Background or Filling, 3. Lace (frame background), 4. Lace Line, 5.Veladura (Dry brush effect), and 6. GE (Generative Element Boundaries). See their effect in the figure. You can see these colors with commands 'u', change the actual Item Color with  'C, and change its main hue with command 'c'. See these and others commands in ColorColor.



The RigeRige and its LatticeLattice, theoretically infinite, are limited either by the display limits or by special boundaries adopting different shapes and borders. These are:

          Square, Rhombus, Door, Circle, Polygon, Star, Arch.

          The sizes of some of these shapes conform with the Lattice and Rige characteristics, to harmonize them with the form inside; but these sizes can also be changed, by means of a factor.

          The shapes can be done by a clean cut of the figure, or be adorned with a border made with the actual frame used in the Rige itself. Type 'p' or use the Menu.

$#+Manual Draw

Perhaps you might want to add some lines or symbols to the figure, to see the effect of a new straight line or make an illustrative sketch. Use then the mouse, Click Left button for points (small circles), Left moving for continuous curved lines, Right moving for straight lines ‒they will be circles instead, if you DoubleClick Left first (please not Right because you would change color); do it again and they will be straight lines once more. Click Right to pick up the pixels ColorsColors to be used in the following drawings ‒you have the Lattice Line color as beginning default. You can also change the Width of each one of these lines, to mark some of them as important. Note that picking up the background color (white or otherwise) you can actually erase previous lines.

          While moving, you will see in the window's caption (where the titles are) the actual values of the point coordinates (for Left button) or both the initial and the present point coordinates, together with the line inclination (angle with the horizontal) and the distance between both points (for the Right button).

          Press a key while the mouse Left button is also pressed, to write the key alphanumeric symbol besides the point or line. Useful symbols, numbers and words can be added in this way in particular points of the figure. The font size can be changed in the window menu (Manual/Font).

             These simple tools allow you to make useful diagrams over your RigeRige or LatticeLattice. With the Frame/Line menu item, you switch between simple lines and frames, similar to the ones used in your actual Rige. In this way it is possible to try the visual effect of new lines, before introducing them as part of some polygonal.

          Also, when the Fill menu item is set (Yes), any shape can filled with the Filling color defined in ColorsColors: simply click within the shape.


                       MOUSE USE IN PUERTRA


Action                         LEFT                                         RIGHT

Click                           Circlet                                       PickUp Color   

Move                          Curve                                        Shape              

Key                             Prints character

Two Clicks                 Fill with Filling Color

DblClick                     Switch Shape: (Straight, Circle, Rectangle, Star, Polygon, Arch...)


These manual modifications only affect the present figure, and are not saved with the Rige parameters. Therefore, grab and print the figure before changing it.



During all the drawing process (manual or automatic) you can change some characteristics of the drawn lines: their Width, with four values ‒default is Min(imal);  the Mode, a  rotating switch of 16 possibilities, to find effects of fantasy while moving the figure around; the Erase possibility (one of the former 16 modes) of seeing the figure transformations with or without erasing the previous situations ‒default is Yes.

$#+Untitled 1


Getting StartedGetting_Started







Generative ElementGenerative_Element

Translation ElementTranslation_Element


Formal RulesFormal_Rules


Plane TransformationsPlane_Transformations

Block DisplacementsBlock_Displacements

Spatial TransformationsSpatial_Transformations

Spatial SurfaceSpatial_Surface



Rige EditionRige_Edition

Lattice EditionLattice_Edition

Gyrated TranslationGyrated_Translation

Circular RepetitionsCircular_Repetitions



Rige AnalysisRige_Analysis

Rige FinishingRige_Finishing



Manual DrawManual_Draw



$ Rige Edition

# Rige_Edition

+ 00001

$ Lattice Edition

# Lattice_Edition

$ Gyrated Translation

# Gyrated_Translation

+ B:00001

$ Circular Repetitions

# Circular_Repetitions

$ Polygonals

# Polygonals

$ Files

# Files

$ Rige Analysis

# Rige_Analysis

+ A:00004

$ Rige Finishing

# Rige_Finishing

$ Background

# Background

$ Colors

# Colors

$ Boundaries

# Boundaries

$ Manual Draw

# Manual_Draw

$ Lines

# Lines

$ Untitled 1

# Untitled_1

$#+K  Menu




See it in Help of DP21_MENU


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