Peter Denies Christ

From the 2003 Cityside Easter Art exhibition, a medieval-Celtic-influenced take on the story of the threefold betrayal. Featured in the (unreleased) documentary The Real Easter Art(ist) Thing by Gwen Strickland and Duncan Philps.

Peter kneels at the centre, weeping, in the moment when the rooster has crowed. He recalls (in the roundels) the moments leading up to this: Christ's prediction (top roundel), and the three betrayals themselves. But woven into the structure of the work are hints of the future moments when he will again affirm rather than deny his master. The fish recall for us the moment by the Sea of Galilee at the end of John's Gospel, and the roundels form an inverted cross, on which Peter died, according to tradition, having asked that he not be executed in the same way as his Lord. The keys are, of course, the traditional Keys of the Kingdom associated with Peter.

Though inspired by medieval Celtic art (particularly the Book of Kells), the painting tells a story, which medieval Celtic art typically did not do. It is laid out according to principles of sacred geometry.

The blond dreadlocks Christ is wearing are an indicator of high status. Important medieval Celts dressed their hair with lime, which bleached and matted it.

Peter Denies Christ

Peter Denies Christ (2003) by Mike Reeves-McMillan. 600 x 850mm, acrylic, gouache and metallic inks on board. Collection of the artist.

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The Digital Rose Window

This is a work of digital medievalism - a "rose window" constructed by ancient geometric principles (in CorelDraw, as it happens), incorporating digital camera images of actual roses instead of stained glass.

Digital rose window

I have written an article (Constructing a Digital Rose Window) which sets out step-by-step how I did created this piece.

After I had made the piece I realized that parts of it were quite "fiery", and therefore suitable for a dedication to the Celtic saint Brigid. I had been thinking about making an icon of Brigid because, not long before, I had performed a ceremony for myself which used the symbolism of Brigid (along with Mary and the archangel Michael, three favourites of the collection of Scottish prayers known as Carmina Gadelica), and had found it significantly healing.

The rose window (or the symbol of a rose generally, particularly if it is white) is often associated with Mary; the roundness also has a feminine association, and my ceremony dealt with my relationship as a male to the feminine.

You could think of the inner section (the three white roses layered on and within one another) as relating to Mary, the fiery squares and trefoils to Brigid (whose triple nature was as smith, poet and healer) and the outer, protecting golden elements to the guardian Michael. The Trinity is, of course, woven in there too in the recurrence of the number three.

I may yet do the icon, but in the meantime, I dedicate the Rose Window in a prayer based on the style of the Carmina Gadelica:

In the light of the Creator,
In the love of Christ,
In the life of the Spirit,
I dedicate this window.

To Mary the Mother,
To Brigid the Healer,
To Michael the Champion,
I dedicate this window.

Under the moon gentle,
Under the sun generous,
May the light shine through me,
And may I blossom like a rose.

Creative Commons License

(Originally created in 2006)

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Constructing the Digital Rose Window

This article outlines my main steps in constructing a digital "rose window".

Medieval rose windows, like all parts of medieval cathedrals (and, indeed, like most places of worship going back as far as the temples of ancient Egypt) were constructed using "sacred geometry" to lay them out. This consists of geometric constructions of various kinds which provide a number of lines and points on the surface which one can use in one's design (rather than just a grid of equal measurements, which is what tends to be used in modern designs). There is always a lot more geometry in the planning stages than in the final design, so "sacred geometry" has three levels: the "hidden" geometry of the construction lines, the "manifest" geometry of the visible design, and the symbolism of the numbers embedded in both of these.

First, I took my digital camera (an ordinary 4-megapixel Nikon Coolpix, not an expensive professional device) to the free, public Parnell Rose Gardens on a nice day when the roses were in bloom. I had a pleasant time taking a few dozen photos to act as "stock".

Some time later, when I had the time, I pulled out the old copies of CorelDraw and Corel PhotoPaint which I had bought for $70 on the TradeMe auction site. You could equally well use different programs, such as Adobe Illustrator and Photoshop if you have them, or for free alternatives Inkscape and The Gimp. I find the Corel products slightly easier to use than the free ones (though they are probably not as easy as the very expensive Adobe programs, or for that matter more up-to-date Corel ones).

I Googled "rose window geometry" and found some helpful stuff. This website helped me construct the outer part of the window, based on the west window at Chartres Cathedral, while this website gave me the idea for the central pattern (adapted somewhat from the pattern given there, which seems to make the central circle an arbitrary size).

Stage 1: Dodecagon

The first stage of my construction, once I had defined the centre of the page using guidelines, was to set up the twelve-pointed star (dodecagon) which is the basis for most of the layout. I drew an arbitrarily-sized starting circle at the centre of the page, then used the Duplicate tool to produce four more circles the same size and moved them so that they met at the centre. These, plus the centred horizontal and vertical guidelines, define the points on the original circle's circumference which are the points of the star:

Stage 2: Outer Circles and Guidelines

The next stage was to place twelve small circles nesting in the arms of the star. I also, for ease of reference, ran guidelines through the twelve points of the star (these don't come through when I export to GIF):

I have started here to use different colours to distinguish the lines from each other so that the whole thing is easier to read. Later on I separated the construction lines onto their own layer in the drawing so that I could hide them to get an idea of what the final piece would look like.

The blue-green circles above are not actually construction lines as such. I drew them to check that all my circles were properly aligned with each other. In CorelDraw, you can draw a circle from a defined centre (such as the crossing point of the construction circles) by holding down Shift and Ctrl while you click and drag from that centre. The other programs I mentioned have similar capabilities. A hint: it is a good idea to turn on "snap to guidelines", since this will help to get your circle's centre aligned with the centre of the design. Consult your program's Help to find out how to do this.

Stage 3: Inner Dodecagon and Outer Boundary

I then drew another dodecagon inside the circle defined by the twelve smaller circles. I constructed an equilateral triangle (out of guidelines) with the uppermost point coinciding with the uppermost point of the larger dodecagon, and within this triangle constructed a circle which cut the small circles. This will be the actual outer edge of the window, the constructive geometry of which therefore extends outside its boundaries. I then (on my "shapes" layer) duplicated the twelve small circles and erased everything outside this boundary. (To erase parts of circles in CorelDraw you have to convert them to curves and then deselect the tick in their properties that says they are closed shapes.)

Stage 4: Squares and Trefoils

Where the dodecagons cross each other they form squares, which are the next part of the design. A few of my construction lines got accidentally moved, as you can see (a drawback to digital work that you don't get on paper):

I substituted trefoils for the quatrefoils in the Chartres window, using the helpful little book Sacred Geometry by Miranda Lundy (Wooden Books, 1998) to find out how to construct them. I used the intersections of the inner dodecagon (shown in orange above) to get the initial (outer) triangle:

From the point illustrated above, you just erase the parts of the circles that are within the innermost triangle and hide the triangles, and you are left with a trefoil. Hint: A quick way to make an equilateral triangle is to draw the first side, create a duplicate rotated 120 degrees, and then create a duplicate of that rotated a further 120 degrees. Then move the sides so that the vertices (corners) touch each other.

Once I had one trefoil, I created rotated duplicates (there are twelve, so each one is rotated 30 degrees from the previous one) and moved them to the appropriate points on the design. To make sure I had them in the right place I drew a couple of circles, a yellow one through the centres and a green one around the outside of the band of trefoils (the yellow one wasn't needed, actually):

Stage 5: Inner Circles

I now drew a series of shapes to give me the central figures. First, a circle passing through some of the inner crossing points of the orange dodecagon. Within this, I constructed a hexagon and (using guidelines, so you can't see it) a square. I discovered, interestingly, that if I drew twelve circles with centre points on the circle constructed within the square, such that they each took up a twelfth of the circle (based on the twelve equally spaced guidelines), they also fitted exactly inside the hexagon. There is no doubt a clever mathematical proof of why this is, but I will leave that as an exercise for the reader:

I erased the parts of the circles inside the inner circle:

I then used one of the standard step-down methods of getting a smaller circle with a harmonious relationship to the larger one, namely, drew a square in the larger circle (with guidelines) and a circle in that square. Then I did the hexagon/square/twelve circles maneuver again at the smaller size:

Here is the completed drawing, with construction lines removed:

Once I had the drawing, I switched over to PhotoPaint. Inserting the photos was relatively straightforward, just a matter of selecting part of the drawing and pasting the appropriate photo (or part of it) into the selection. It took a while, partly because I wasn't totally familiar with the tools and at one point locked a layer accidentally, which prevented pasting into that layer. But apart from this it went fairly easily.

Here is the final result (a larger version can be seen here):

I'm not a trained artist or even much of a geometer - I can just about follow the proofs in Robert Lawlor's fascinating book Sacred Geometry (Thames and Hudson, 1982) if I concentrate hard, but I prefer Miranda Lundy's simple diagrams to help me do constructions. (If you can't get hold of either book, try Euclid and Archimedes for ideas about geometric constructions, or Google for tutorials on sacred geometry. I have also written an article on Digital Sacred Geometry myself.) I created this piece using inexpensive tools which I taught myself to use, and photos I took myself. There are free tools, and even free stock photographs, available to everyone on the Internet. There's no reason why, with some patience and imagination, you couldn't do something like this too.

(Originally written in 2006)

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Digital Sacred Geometry

"Sacred geometry" is the art and science of using geometric figures to design sacred architecture or artworks. Recognising that mathematics is something that humans discover - we don't invent it - and that it corresponds to forms found in nature, to aspects of music, and to cosmological measurements, ancient peoples beginning (as far as we know) with the Egyptians developed the use of geometric forms to lay out their temples and altars. The Greeks further developed geometry in both its sacred and pragmatic aspects. Pythagoras, known for his theorem of the right-angled triangle, was the leader of a mystical religious movement. Though he may (or for all we know, may not) have developed the formal theorem of the right-angled triangle, he was not the first to note its special properties, a triangle with sides of three, four and five units having long been used in Egyptian art.

In the Middle Ages and Renaissance, sacred geometry was used to lay out cathedrals, at all levels of detail from the floor plan to the fine decoration and the composition of the paintings. The sense of harmony and proportion one experiences in a medieval cathedral (or a Greek temple) is not only because of the religious associations of the location, but because geometric principles have been used to lay it out. In contrast, modern architecture is often designed based on a simple grid, and as such does not have the same aesthetic appeal. Although the underlying geometry on which the layout is based is not evident without careful analysis, as humans we seem to respond to the proportions of a geometrically planned building or artwork with pleasure, as we do to a natural scene.

Nature is also full of geometry, from the hexagonal cell of a honeycomb to the spiral of a nautilus shell, and there are excellent reasons of resource optimisation and stability for adopting these structures.

I am interested in creating artworks based on sacred geometry, and indeed on creating religious artworks which refer to the medieval tradition. I became interested initially when I took a class on Celtic design, in which a complex tradition of sacred geometry is embedded, seen both in the pagan works of the La Teine and Hallstadt cultures and in the magnificent illuminated manuscripts of early medieval monks. I have, however, some resources which those early artists lacked, in the form of digital drawing tools, and this guide specifically focusses on the use of such tools in sacred geometry.

Digital Tools

For budgetary reasons I don't own a copy of Adobe Illustrator, which is probably the premium software for this kind of work (it costs about $500 US). I do have an old copy of CorelDraw version 7, and have recently downloaded a copy of Inkscape, a free software vector graphics program. "Vector graphics programs" are what these various tools are called, meaning, broadly, programs with which you can draw pictures based on lines with a mathematical description, rather than on tiny coloured squares which are described individually (known as bitmap graphics).

One implication of this is that you can scale the pictures up or down without losing detail. Another, of course, is that they, too, are using geometry underneath, and so are entirely suitable for the task at hand.

The traditional tools of the sacred geometer are the compass (for drawing circles) and the straightedge (for drawing lines). From these two elements - circles and lines - they could draw everything they wanted, except of course illustrations of things in the real world, which they drew freehand within the spaces defined by the circles and lines. In theory, then, the only tools we need to use in our digital drawing programs are the circle tool (or ellipse tool constrained to a circle) and the line tool. If you are a purist you can stick to those. Personally I take a few shortcuts, as below.

I also have a copy of the free geometry program KSEG. KSEG is specifically designed for "compass and straightedge" geometry, and is quick to use, but quite basic (and a little quirky until you get used to it). The examples in this article are done in KSEG. I'm not sure that I'd use it for art, though, probably just to play around with the constructions. Apart from anything else it puts visible points everywhere, which is useful for explaining constructions but not for artwork, where they would need to be removed.

I won't be giving elaborate screenshot-laden tutorials about using the specific programs, since I don't have Illustrator and my version of CorelDraw is old. Anything you need to know how to do you should be able to look up in the Help, and generally the programs work much the same as each other at the basic level. You select which tool you are currently using by clicking it in a toolbar; it then changes the mouse cursor and also the behaviour you see when you click, drag, double-click, etc.

The selection tool (an arrow) selects individual items on the screen which you can then manipulate in other ways, moving them, changing their properties (like colour or line thickness) and so forth. You will need to right-click or use the menus to get to some of these options.

The ellipse tool draws ellipses when you click and drag. You can (and should) "constrain" the ellipse to be a circle by holding down a key (Ctrl in CorelDraw and Inkscape, Shift in Illustrator apparently), and also make it expand out from the point you first clicked (by holding down Shift in CorelDraw or Inkscape, Ctrl in Illustrator as you drag the mouse). This is the behaviour you will usually want from your circles, and it is equivalent to placing one leg of your compass at that point (click) and positioning the other leg (drag). With a compass you would then have to rotate it to draw the circle; the digital tool does this for you automatically.

In Inkscape, you will need to set fill to "No paint" and stroke to "flat colour" in the Properties dialog (Object > Fill and stroke on the menu). Otherwise you will get filled-in circles instead of the outlines you want.

The freehand drawing tool (pencil/pen) will draw a line between two points that you click, and is the equivalent of the straightedge.

The other main tools I use are the symmetry tools, particularly rotation, since I often base my pieces on a circular shape and want to have smaller elements distributed evenly around the circle. As well as rotating you can also reflect or flip elements (horizontally or vertically). In CorelDraw at least, you have the option to apply the transformation to a duplicate (or copy) of the object, and this can be very useful because it leaves the original in place and moves the duplicate.

I use rotation with duplication as a shortcut when creating regular polygons such as squares or equilateral triangles. Draw the first line where it belongs, duplicate and rotate a number of degrees equal to 360 divided by the number of sides (e.g. 120° for a triangle), and repeat until you have the number of sides you require. Then move them (keeping the same angles) until their ends touch. This is a foolproof way of constructing regular figures and is a lot faster than doing it the purist way with lines and circles. There is a purist way to do this for any given figure of a reasonable number of sides, but some of them are quite arcane (a seven-sided figure is particularly difficult, and the result is imperfect).

Also very useful are guidelines. You can drag these from the horizontal and vertical ruler bars (which I mostly ignore otherwise, since my constructions are based on geometry, not measurement). They look like dotted blue lines, and help you to line things up with each other and save you cluttering your layout with drawn lines which you will later have to remove. You can have angled or sloping guidelines as well as horizontal and vertical ones, except in Inkscape (as far as I can see), in which you will have to use drawn lines. (You can lock these to prevent them from being accidentally moved - right-click and select Object properties, then turn the Lock checkbox on. However, I can't figure out how to unlock them if you need to - perhaps you can't.) I believe later versions of CorelDraw (and probably Illustrator) may allow circular guidelines as well, and I am planning to propose to the Inkscape developers that they implement both angled and circular guidelines.

You should turn "snap to guidelines" on in your program if it has this option so that when you click near a guideline the mouse cursor will automatically move slightly to be over the guideline - this makes for greater accuracy.

It is a good idea to use multiple layers in your drawing, one to hold the construction lines (which won't appear in the final piece) and one to hold the "manifest" lines, which will appear. You can copy or move things from one layer to another and hide the construction line layer to check your progress. You may have to delete the construction lines before exporting the picture to another format, because even if you hide them they will appear when exported in some versions of some programs.

Sometimes you will only want part of a shape, and you will need to erase the rest of it using the eraser tool. In CorelDraw (at least the version I have), if the object is a circle you need to convert it to curves first (Ctrl + Q), and then go into its properties (right-click) and untick the checkbox which says to close the curve, otherwise the erasing won't work.

You can also use commands to remove areas where two objects overlap (Trim in CorelDraw), to join them together (Weld), or to create another object from the overlapping portion (Intersection). Some programs (such as Illustrator and Inkscape) also enable you to break lines apart at the point where they overlap, after which you can delete one part of the line. See Inkscape's Path menu and its Help.

That's about it as far as using the tools goes. The rest of this guide covers the basics of using lines and circles to draw more complicated shapes.

The Mandorla

The mandorla (meaning "almond"; it's also known as a "vesica piscis", or fish's liver) is the almond or fish-liver shape you get when you overlap two equal-sized circles in such a way that the centre of each is on the other's circumference:

The mandorla was much used in medieval art, often containing Christ enthroned or the Virgin and Child. The symbolism is multiple. The shape is like a stylised fish, Greek "IChThUS" (ΙΧΘΥΣ), which from early times was used as an acronym for "Iesous Christos Theou Uios Soter" - Jesus Christ, God's Son, Saviour. It also represents the meeting of two worlds (heaven and earth) in the person of Christ. The shape also resembles that of an eye, and it is used as such in art, though mainly in Muslim rather than Christian contexts from the examples I have seen. Finally, it can be seen as lips (by which God utters the Word) or as female genitals (by which Christ enters the world).

The relationship of the width of the mandorla to its height is the proportion 1:√3, a ratio which turns up in a number of other places in sacred geometry.

A mandorla can be used to construct several equilateral triangles:

It can be used to construct a rectangle in the proportion 1:√3:

Or a rectangle with a diagonal which is √5:

It can also be used to construct a hexagon:

With the help of a couple more circles, you can construct a pentagon on it:

All of the above constructions are sourced from Robert Lawlor's book Sacred Geometry (see below for bibliography).

The In-Triangle and Out-Triangle

An in-triangle is an equilateral triangle which fits exactly in a circle; an out-triangle is an equilateral triangle in which a circle exactly fits. The in-circle of a triangle is exactly half the size of its out-circle, usefully.

You can draw an in-triangle with the help of one additional circle:

To draw an out-triangle is very easy with angled guidelines where you can set the number of degrees - just use guidelines at 120° from each other and make them tangents of the circle (that is, make them each contact the circle at just one point on its circumference). Here is a slightly more complicated way which requires you to work out the in-triangle and bisect two of its sides (see below for how to bisect a line segment):

The In-Square and Out-Square

The in-square and out-square work similarly to the in-triangle and out-triangle.

Probably the easiest way to draw an in-square is to put horizontal and vertical guidelines through the centre of the circle, connect the points at which they cut the circle, and rotate it 45° if required:

An out-square is also easy with horizontal and vertical guidelines which are tangent to the top, bottom and two sides of the circle.

The Pentagon

We have already looked at drawing a pentagon from a mandorla. Here are two ways to draw a pentagon inside an existing circle, the first from Lawlor (p. 36), the second from Lundy (pp. 26-27, see bibliography). In the Lawlor one, draw the lines and circle shown in red first, then the blue circle, then the green circles. In the Lundy one, draw the red, blue, green and brown circles in that order.

My books don't tell me in detail how to draw an out-pentagon, so I have had to come up with my own (cheat's) way, inspired by a diagram in Lawlor, p. 53. Draw an in-pentagon as above, then arrange guides which are tangent to your circle at the points where the in-pentagon touches it. (The guides will be at 72° intervals, since five of them make up the 360° of a circle.) You actually only need to place three (adjacent) guides, which will give you the length of one side by drawing a line between the crossing points of the guides. You can then duplicate and rotate that side five times, rotating 72° each time, and move the resulting lines until their end points touch and they form a pentagon.

Alternatively you could adapt the approach given above for the out-triangle (bisecting the sides, drawing a line through each vertex and the middle of the opposite side, and placing further lines which cross those at right angles where they meet the circumference of the circle).

Bisecting a Line Segment

We don't need to look at drawing hexagons - we covered in-hexagons in the mandorla section, and you can apply the same approach to drawing out-hexagons as I gave above under pentagons. So let's move on to a useful technique of bisecting a line segment, which I mentioned above under the triangle.

First, draw a circle of arbitrary radius (but with the radius more than half of the length of the line segment), centred on one end of the line segment.

Duplicate the circle and move the duplicate so that it is centred on the other end of the line segment.

The circles will cross at two points, and a line drawn (or a guideline created) between those two points will cross the original line segment at its exact centre:

Trisecting a Line Segment

Sometimes you want a third rather than a half. Here are a couple of different geometric ways to get a third.

The first technique (Lundy, p. 20) is to draw a hexagram (Star of David) in a circle and another hexagram inside it. The circle constructed through the inner crossing points of the inner hexagram is one-third of the out-circle of the outer hexagram. (To draw the hexagram, create an in-triangle and invert a duplicate of it.)

The other technique (Lundy, p. 32) involves dividing a square into nine squares. Use the bisecting technique or guidelines to get the centre of each side of the square, and draw lines from the centre of each side to the two corners of the opposite side:

The four innermost crossing points divide the square into nine (and each side of it, therefore, into three):

The middle four crossing points divide it into sixteen (and the sides into four):

And the outer eight crossing points divide the square into twenty-five (and the sides into five, if you were wondering how to divide a line into fifths):

Packing Circles

The effect of packing circles into a geometric figure so that each one touches the next at only one point and they all touch the outside of the figure is a pleasing one. The simplest packing is six circles around a central circle, which you can do with circles of any size using this construction (Lundy, p. 9):

Draw the red circles first, then the blue circles, then the green lines to give you the centres for the outer circles.

If you wanted to pack the circles inside an existing circle, see the first technique under Trisecting a Line, above. You could use the same approach to pack them inside an existing hexagon by first drawing the hexagon's in-circle.

Is there a general solution for packing circles inside an existing circle, though? Yes, there is. Lundy (p. 46), as is her way, gives several illustrations from which you can figure out the principle, without explaining it in her text.

Draw an out-polygon around the circle, with the same number of sides as you want circles. (A triangle for three circles, square for four and so on.)

Draw guidelines through the centre from the points (vertices) of the polygon and also from halfway along each of its sides (bisecting the sides as covered under Bisecting a Line, above).

The centres of the packed circles will be on one set of lines (either the ones from the vertices or the ones from the halfway marks), and the circumferences will be on the other set. Call them Set A and Set B.

The tricky part is finding at what point on the lines the centres of the circles fall. To find this, you will need to construct a right-angled triangle, one side of which is formed by a vertical or horizontal line from either Set A or Set B, and the long side of which is formed by the immediately adjacent line from the other set.

You will then need to bisect the angle formed by the long side and the side which is at right angles to the vertical or horizontal line. The line constructed on this bisected angle cuts the vertical or horizontal line at the point where the circle's centre should be placed.

I'm beginning to see why Miranda Lundy didn't try to explain this - a diagram makes it much clearer. The red dotted lines are Set A and the black dotted lines Set B. The blue dashed lines outline the equilateral triangle, and the green dotted line is the bisector of the angle. Not shown is the construction circle you would probably draw from the figure's centre through the first two centres you find (the ones on the green line), which would then give you the remaining centres.

To bisect an angle is one of Euclid's early theorems. You can, of course, cheat and use your digital tools to measure the angle, then divide it by two (or if you're using KSEG, it has a "bisect angle" button). But if you want a more classical method, here is my suggestion (I don't have Euclid to hand right now, so I figured this one out from first principles).

First, make a sub-triangle which has sides of the same length on either side of the angle you want to bisect.

You can do this by drawing a circle of arbitrary size centred on the vertex where the angle to be bisected is, cutting both of the sides of the triangle which meet to form that angle.

Join the two points where the circle cuts the sides.

Now bisect the line you have just drawn using the method given under Bisecting a Line, above.

A line drawn from the vertex of the triangle through the midpoint of the opposite side will bisect the angle:

Note that the bisecting angle will not necessarily pass through the midpoint of the opposite side - this only occurs in special cases. So you can't use that as a shortcut.

Once you have your packed circles, you can create pleasing lobed shapes by erasing the parts of the circles which fall inside the inner geometric figure (or using the Trim command in CorelDraw or its equivalent in other programs to remove the area where the objects overlap). See my article on Constructing a Digital Rose Window for an example - in that case the figure is a trefoil (three lobes), but you can use the same principle to create quatrefoils, cinquefoils and so forth.

I wondered to myself: Is there a general method of packing circles in a given polygon, not just in a given circle? After playing around for quite a while I hadn't found a general method (which isn't to say there isn't one), but I had found several specific methods for particular polygons. Basically you need to find a point which is equidistant from the outer boundary of the polygon and from the lines which pass through both its vertices and its centre, and which is not closer to the centre than it is to the boundary. Here is my first solution, for a hexagon, which being made up of equilateral triangles is easy:

My second, for a square (also easy):

My third, for a triangle. This was a lot harder and involved several hours of experimentation. In the end I hit upon constructing a square on the diagonal from the centre to the lower left vertex (square shown in blue). The centre of the square (found by the green lines) is a point on the circle (brown) which, where it crosses the red guides, provides the centres of the packed circles. Whew!

I was going to leave the pentagon as an "exercise for the reader" - which is code for being too lazy to work it out - but I had got fascinated by this time and attempted the pentagon too. I worked these out, by the way, starting with the end state of a number of packed circles, drawing the polygon around them, and then trying to work backwards to find out how, given the polygon, I could have drawn the circles.

So my first task, having drawn circles around the vertices of a pentagon, was to draw the larger pentagon they all fitted into. I'll record here how I did this (basically by constructing a square one circle-radius in width on several of the key points to give the larger pentagon):

Now that I had my pentagon packed with circles, I could work backwards and attempt to find a construction which would give me the centres of the circles. Since a square helped me last time, I decided to go for the most obvious square (actually rectangle) - the one built on the pentagon's baseline - and voila:

A circle the width of one of the pentagon's sides, centred on the pentagon's centre, gives the centres of the circles which pack that pentagon.

Sadly, this is not a general solution - it only appears to work for pentagons (as you will see if you try with the hexagon). If there is a general solution to the problem of fitting circles (equal to the number of sides) into a polygon, it has eluded me. Of course, I am completely ignorant of most of the work ever done in geometry, so the solution may well have been found centuries ago.

Drawing Significant Rectangles

There are several rectangles which are obviously well-proportioned to human eyes and which make a good basis for laying out your artworks. The first is the shape of the A series of paper sizes (A4 being the best known). This rectangle has the interesting property that when you cut it in half across the middle of the long sides, each half has the same proportions as the original. So an A5 sheet, for example, is half the area of an A4 sheet but has the same ratio of height to width (making the A series useful for reducing and enlarging photocopies).

The A series is a √2 rectangle (if you call the width 1, the height is √2 or approximately 1.414). It is easy to construct, because this is the proportion of the width of a square to its diagonal:

A circle with radius equal to the diagonal of your starting square gives you the √2 distance to construct your rectangle. The red portion is the area added to the original square to give the rectangle.

We already gave the mandorla construction of the √3 rectangle, but here is another way to do it (based on Mackinder, p. 14, see bibliography):

Start with a square divided into four. Draw the red circle first to give a √2 rectangle, then draw the blue circle which enlarges it to √3. The green portion is the area added to the original square to give the rectangle.

The so-called "golden rectangle" has the property that if you remove a square based on the shorter side, what remains is a golden rectangle. It is based on the distance from the middle of one side of a square to one of the opposite vertices (corners):

The blue portion is the area added to the original square to give the rectangle. This is a particularly easy construction.

The Golden Rectangle is supposed to be particularly aesthetically significant, besides being the sort of thing that someone like Lawlor gets very excited about because of its many unusual mathematical properties. It is based on the "golden proportion", known by the Greek letter phi (φ). Its unique property is that the ratio 1: φ is the same as the ratio φ:1+ φ. It is approximately 1.618, and exactly (√5 + 1)/2.

Building Up a Layout

Once you have your base shape, whether square or rectangular, you can start drawing layout lines.

Start by getting the midpoints of each side (see Bisecting a Line, above). Draw vertical and horizontal guidelines which divide the rectangle or square into four equal parts.

If your layout is rectangular, construct squares based on the short sides (you can use a circle with its centre at one corner and its radius equal to the short side; where it cuts the long side is another corner of the square). Depending on the exact shape of the rectangle these will overlap or fall short of overlapping, giving you another rectangular space between them and two more points on the long sides.

Draw horizontals or verticals which divide the squares in half in whichever direction they are not already divided.

Join up all the points you now have on the outside of your rectangle, diagonally, in all the combinations you can think of. Join them to the centre as well.

The crossing points of these various diagonals (and the horizontal and vertical lines defining the halves of the squares and rectangles) will give you the basic points of your layout, from which you can proceed to draw concentric geometric forms, circles and what have you.

Two crossing points close together can be the basis for a measurement which can be used elsewhere in the piece, or to define the width of a border.

This is how the ancients did it, as you can prove by getting an accurate photograph of an ancient building or an artwork such as a page of the Book of Kells and drawing squares and diagonals. Many of the crossing points will be at significant points of the design (and most of the others are probably based on some more subtle geometry).

Conclusion

Those are the basics (as I see them) of digital sacred geometry. I haven't covered spirals, fractals, sine curves etc.; maybe I will look at these later, when I have made some art with them.

For a worked example of creating a piece of digital sacred geometric art, have a look at my other article on Constructing a Digital Rose Window.

Bibliography

Lawlor, Robert Sacred Geometry (Illustrated Library of Sacred Imagination) (Thames and Hudson, 1982). Considerably more technical and theoretical than what I have presented here. Lawlor presents extensive material on both the "sacred" and "geometry" aspects, with illustrations from art, architecture and natural forms, as well as "workbooks" in which he presents the geometric and mathematical relationships between various figures.

Lundy, Miranda Sacred Geometry (Wooden Books, 1998). A much less technical and more practical book, small, but packed with useful constructions.

Mackinder, Jack Celtic Design and Ornament for Calligraphers (Thames and Hudson, 1999). As it was Jack's night class which first turned me on to sacred geometry, I want to give his book a boost. It's particularly strong on layout but covers all the basic aspects of Celtic design (knotwork, spirals, step patterns, key patterns, zoomorphics).

(All those links are Amazon affiliate links. If you buy, I get a tiny commission.)

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A Trinitarian Rosary

Some time ago, I was reading a book called Pray Your Way by Bruce Duncan, which took the approach of applying Myers-Briggs personality theory to helping people to find ways to pray which would suit their personalities. Out of that came the idea of using prayer beads to help me concentrate and keep my mind from wandering so much. My initial approach was to associate each bead with a different point to meditate on and use the beads as
"aides-memoir", because, as I said at the time, I was "Protestantly suspicious of ‘vain repetition’."

It turned out I was missing the point (as well as setting myself an unnecessarily difficult goal).

The repetition of the prayers in the rosary is not "so that you will be heard by the multitude of your prayers"; it is to lull your chattering mind into a meditative state. The idea is similar to the mantras used by Hindus or Buddhists, or to the Desert Father John Cassian’s idea of the repetition of a short phrase in meditative prayer (in his Conferences). The Desert Fathers were probably the first Christians to use something like prayer beads, in their case made from knotted ropes.

Not surprisingly, I didn’t find these sprawling meditations sustainable and soon stopped using the beads. But more recently, having in the meantime discovered the meditative theory just mentioned, I again tried using prayer beads with much better results.

I am still a Protestant, and can’t quite bring myself to pray the classic Catholic rosary, which involves addressing Mary. I have great respect for Mary, but not so much that I want to pray to (or even through) her.

I have read several Catholic justifications of the practice but I’m just not convinced. So I have developed a "non-Mary-hailing" rosary – assembled, however, from pieces of the Catholic Mass, because I do value Christian liturgical tradition.

I have fitted it to a bead string (or chaplet) of my own invention, the "Trinity" beads. With some adaptation it could be fitted to the traditional rosary or to other chaplets such as the Anglican rosary. Briefly, the Trinity Chaplet resulted from a request by my wife (then my fiancee) for a set of beads which she could use to meditate on the nine Fruit of the Holy Spirit (Galatians 5:22-23).

Trinity beadsThe 33 beads represent the years of Christ’s earthly life (as with the Anglican rosary), but they are grouped in threes and nines,  traditionally sacred numbers
in many cultures. The blue represents the Father (in Heaven), the red the Son
(by association with blood), and the green the Spirit (who gives new life) – this is my own invention and not from any tradition I know of.

The string consists of three "introit" beads (hanging outside the main loop); three "Trinity" beads (the darkest beads in the illustration); and three sets of nine coloured beads, separated by the Trinity beads.

When constructing the string, begin with the Trinity bead which is adjacent to the introit beads. String the coloured and Trinity beads, then feed the string back through the first bead in the opposite direction, and knot the two ends under the bead. Then feed the doubled string through the introit beads and tie it off in a knot large enough to keep them from slipping off. Bead string can often be melted (with a candle or match, used carefully) to prevent fraying.

It would be appropriate to add a symbol of the Trinity, such as a triangle, triskele (triple spiral), or trefoil, below the introit beads in the position occupied by a cross or crucifix in other rosaries. I haven't done so in the string illustrated.

The sets of prayers I have composed are also very Trinitarian, and several of the prayers which make them up break naturally into three parts. This gives a nice rhythm which helps with the "lulling" effect as well as symbolically underlining the theme.

I have also come up with a way of using the prayers without the beads, while driving. (It’s perfectly safe as long as you keep enough of your attention for the road, and it certainly helps to avoid "road rage".) The places where it is easy to lose track of where you are are in the long runs of nine coloured beads between the black beads, so when I am praying these parts I keep track using the middle three fingers of each hand. I count the individual prayers on the left hand and the groups on the right hand.

So, for example, when I am praying the first group of three, I keep my right forefinger slightly tensed to remind me that I am on the first group, and as I pray the three prayers I tense first the ring finger, then the middle finger, and then the forefinger on my left hand. Then when I shift to the second group I tense the middle finger of the right hand, and so on. I have found this works for me quite well; your mileage, so to speak, may vary.

Catholics who pray the rosary often use meditation sequences, the best-known being the Joyful, Sorrowful and Glorious Mysteries.

Reverting to the origins of the Trinity Beads, I have tried meditating on the nine fruit of the Spirit while praying my rosary, and found that it worked well. However, I wouldn’t recommend that you do that when first starting out unless you have done similar meditation work before; go through with just the prayers a few times first to get used to them before adding elaborations.

What benefits have I obtained? I find myself, for the first time in a long time, with a positive emotional relationship to expressions of orthodox faith. Not that I didn’t hold an orthodox faith (though I am orthodox in rather an unusual mode), but that the traditional ways of expressing this held some negative (and no positive) emotional charge for me. Praying the rosary has changed that for the better. I also find it calms my mood overall, which is good on the way to work (and occasionally on the way home, depending what has happened).

The Prayers

Being me, I first of all came up with something wordy and complicated which took half an hour. Here is a much simpler one which takes about five minutes. It uses gender-neutral" language for God. You can restore the words "Father" and "Son" in place of "Creator" and "Redeemer" if you are more comfortable with that, of course, or go in the other direction and find an alternative to "Lord God Almighty".

On each of the three black "introit" beads:

Holy, holy, holy, Lord God Almighty; heaven and earth are filled with your glory; hosanna in the highest.

On each round black bead:

Glory be to the Creator, and to the Redeemer, and to the Holy Spirit [of wisdom],

As it was in the beginning, is now, and ever more shall be,

World without end, amen.

On each coloured bead:

Creator, have mercy on us;

Redeemer, hear our prayer;

Holy Spirit, grant us peace.

This leaves plenty of room for constructing meditations or prayers in groups of three or nine. Here are some suggestions:

  • The nine Fruit of the Spirit (Love, Joy, Peace, Patience, Kindness, Goodness, Faithfulness, Gentleness and Self-Control). I rearrange them slightly from the biblical order, swapping Patience and Gentleness, because the groupings of three are then more clearly themed.You could either use three beads for each one and go around once, or nine beads for each and go around three times.
  • God, Humanity and the Creation.
  • Christ as Prophet, Priest and King.
  • The person of Christ, the teaching of Christ and the
    people (or community) of Christ.
  • Relationship with God, others and self.

I have taken to using an example of the last, as follows:

Black bead 1: Father, draw us closer to you, as you are doing. I give my heart and will to this.

Black bead 2: Jesus, draw us closer to you and to one another, as you are doing. I give my heart and will to this.

Black bead 3: Holy Spirit, draw us closer to you, to one another, and to our true selves, as you are doing. I give my heart and will to this.

You can download an MP3 recording of me saying the shorter form, with original musical backing. It's 4.5MB, 4 minutes 45 seconds.

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St Margaret’s Church Shop

Near where I lived at the time I wrote this (2002), there's an opportunity shop called St Margaret's Church Shop. Every time I saw the name, bits of dialog like the following started running through my head - in the voices of the Pythons, especially John Cleese. Obviously deeply significant...

Scene: St Margaret's Church Shop. Assistant is behind
counter.

Bell rings. Enter Customer. Looks around nervously.

Assistant Can I help you, sir?
Customer Oh - ah - yes. I'm looking for a church. Are you St Margaret?
Assistant No, sir, I'm her assistant. She has the day off on Thursdays.
What were you looking for exactly?
Customer Well, I don't know. What have you got?
Assistant Is it for yourself, sir, or as a gift?
Customer Oh, for myself.
Assistant Yes, most people wouldn't take one as a gift. Well then, the Anglican here is one of our most popular models. Classic styling - been around a long time. Very flexible, wide range of styles - a wide variety of people like it, and they often find it very comfortable.
Customer Well - it's a little fancier than I -
Assistant Oh, you'd prefer something simpler? How about the Evangelical? Enjoyed wide popularity most of the last century. Very durable, hard-wearing, no-frills kind of church. Were you thinking denominational or non-denominational?
Customer Oh, I don't really mind.
Assistant Well, the Baptist is a popular model. Straight, conservative,
a very respectable brand of church.
Customer Well, perhaps that's a bit too...
Assistant Oh, were you looking for something a little more adventurous?
Customer Well - yes. I mean, not to excess, of course, but...
Assistant I quite understand. More recently they've started making the
Baptist model in Charismatic. Would that interest you at all?
Customer Hmmmm. I have heard that the Charismatic is a little - well -
unstable.
Assistant Yes, I have to admit that they have been known to split
occasionally, but of course our warranty would cover that - and really, they're making them much better these days. Really reduced the number of defects. Oh, except for the ones
with the Toronto Blessing option, of course. I wouldn't say this to everyone, but you clearly like a more conservative church, and the TBs are - well, they just haven't got the design right yet.
Customer Yes, I'd heard that.
Assistant Or there's this new line. Just coming into fashion. The
Post-Evangelical. Not everybody's church, but for those who are starting to find the Charismatic a bit dated - after all, it was originally a 1970s design...
Customer What features does that one have?
Assistant They're still trying out various ones - it hasn't settled
down yet. Liturgy - do you like liturgy?
Customer Weeell...
Assistant They don't all have liturgy. They mostly have the arts,
though.
Customer I can't say I really...
Assistant And then there's spirituality. A sense of exploration and
journeying. Questioning, without necessarily seeking after definite answers. Living life in a permanent state of uncertainty...
Customer I don't think I should like that at all.
Assistant No, no. I daresay you wouldn't. So, where did we get to?
Customer You were showing me the Charismatic.
Assistant Oh, yes, the Charismatic. Really very mainstream now, you
know. Don't be put off by its resemblance to the Pentecostal.
Customer Oh, I know better than that.
Assistant Good. Though the Pentecostal is becoming very mainstream too,
and if you wanted to look at a few I've got some out the back...
Customer No, no, don't bother. I'll take this one.
Assistant Thank you, sir. Cash or charge?
Customer Oh, charge, please.
Assistant There you are, sir. Have a good eternity.
Customer You too.

Exit Customer.

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27 Haiku on the Gospel of Thomas

A friend gave me the Gospel of Thomas (early non-canonical gospel) to read, and it just cried out to be made into haiku.

I wrote the original 21 haiku one evening (1 August 2002) while reading through Craig Schenk's version of the Gospel of Thomas. Originally not all of them were "classic" haiku (5-7-5 syllables), but I later (February 2005) revised them so they all were, and added another six for a new total of 27. Some were extensively rewritten, others just had the line breaks shifted and a couple needed no changes. The six new haiku are numbers 6, 7, 9, 10, 23 and 25.

They're probably not great haiku but the subject matter gives them a boost.

More about the Gospel of Thomas here. That site also has a FAQ.

The first three haiku are based on the translation of the Greek version, the others are from the translation of the Coptic version. I'd like to learn Coptic and read it in the original, but I don't have the energy just now. Verse references are to the Coptic version.

1. (v.2)

Seek until you find
Finding, wonder; wondering, reign,
Reigning, you shall rest.

2. (v.3)

If you will not know
Yourselves, in poverty you
Dwell - and it is you.

3. (v.5)

Recognise what is
Plain before your face; hidden
Things are then revealed.

4. (v.6b)

Tell no lies, and do
Nothing that you hate; Heaven
Sees all things plainly.

5. (v.10)

I have cast the fire
Upon the world, and guard it
Until it blazes.

6. (v. 13)

Jesus is like - what?
Angel? Philosopher? Or
Can your mouth not say?

7. (v. 17)

What I give you, no
Ear has heard, eye has seen, hand
Has touched, or mind grasped.

8. (v.18)

Disciples, who ask
"What will be our end?" - Have you
Found the beginning?

9. (v. 19)

If you heed my words,
And become my disciples
These stones will serve you.

10.  (v. 22)

When two are one, and
Inside is like outside, you
Enter my kingdom.

11. (v.24)

Light is in a man
Of light, lighting up the world -
That is where I am.

12. (v.25)

Jesus said, Love your
Brother - love him as your soul.
Guard him like your eye.

13. (v.28)

I came in the flesh.
I found all of them drunk, and
None of them thirsty.

14. (v.37)

When shall we see you?
When you strip like small children,
Without shame or fear.

15. (v.48)

If two make peace here
In this one house, they can say,
"Mountain, move away."

16. (v.50a)

Whence are you? Of light.
Are you the light? No, but you
Are the light's children.

17. (v. 50b)

What is the sign of
Your Father in you? It is
Movement and repose.

18. (v.58)

Jesus said, "Blessed
Is the one who has suffered
And in it found life."

19. (v.69)

Blessed, those within
Themselves persecuted - know
Truly the Father.

20. (v.70)

Bring forth from within
What will save you - or, if not
Brought forth, will destroy.

21. (v.77)

Split a piece of wood
And I am there. Lift a stone,
And you will find me.

22. (v.82)

Those near to me are
Near the fire - those far from me
Far from the kingdom.

23. (v. 83)
The Father will be
Manifest, but his image
Is concealed by light.

24. (v.91)

You read the sky and
The earth, but do not know him,
The one before you.

25 (v. 106)

When you make two one
You will become sons of man,
And move the mountain.

26. (v.108)

Drink from my mouth - be
Like me; I will become you
And reveal mysteries.

27. (v.111)

Does not Jesus say,
"Whoever finds himself is
Better than the world?"

1. (v.2)

Seek until you find
Finding, wonder; wondering, reign,
Reigning, you shall rest.

2. (v.3)

If you will not know
Yourselves, in poverty you
Dwell - and it is you.

3. (v.5)

Recognise what is
Plain before your face; hidden
Things are then revealed.

4. (v.6b)

Tell no lies, and do
Nothing that you hate; Heaven
Sees all things plainly.

5. (v.10)

I have cast the fire
Upon the world, and guard it
Until it blazes.

6. (v. 13)

Jesus is like - what?
Angel? Philosopher? Or
Can your mouth not say?

7. (v. 17)

What I give you, no
Ear has heard, eye has seen, hand
Has touched, or mind grasped.

8. (v.18)

Disciples, who ask
"What will be our end?" - Have you
Found the beginning?

9. (v. 19)

If you heed my words,
And become my disciples
These stones will serve you.

10.  (v. 22)

When two are one, and
Inside is like outside, you
Enter my kingdom.

11. (v.24)

Light is in a man
Of light, lighting up the world -
That is where I am.

12. (v.25)

Jesus said, Love your
Brother - love him as your soul.
Guard him like your eye.

13. (v.28)

I came in the flesh.
I found all of them drunk, and
None of them thirsty.

14. (v.37)

When shall we see you?
When you strip like small children,
Without shame or fear.

15. (v.48)

If two make peace here
In this one house, they can say,
"Mountain, move away."

16. (v.50a)

Whence are you? Of light.
Are you the light? No, but you
Are the light's children.

17. (v. 50b)

What is the sign of
Your Father in you? It is
Movement and repose.

18. (v.58)

Jesus said, "Blessed
Is the one who has suffered
And in it found life."

19. (v.69)

Blessed, those within
Themselves persecuted - know
Truly the Father.

20. (v.70)

Bring forth from within
What will save you - or, if not
Brought forth, will destroy.

21. (v.77)

Split a piece of wood
And I am there. Lift a stone,
And you will find me.

22. (v.82)

Those near to me are
Near the fire - those far from me
Far from the kingdom.

23. (v. 83)
The Father will be
Manifest, but his image
Is concealed by light.

24. (v.91)

You read the sky and
The earth, but do not know him,
The one before you.

25 (v. 106)

When you make two one
You will become sons of man,
And move the mountain.

26. (v.108)

Drink from my mouth - be
Like me; I will become you
And reveal mysteries.

27. (v.111)

Does not Jesus say,
"Whoever finds himself is
Better than the world?"

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