## Mathematical observations

There are many types of knotwork in various religions or groups. However, below are the observations that distinguished celtic knots from others:

1) The strong diagonals in celtic knots are based on the 3:4:5 triangle, otherwise known as the right-angle triangle.

2) At almost every corner of any celtic knots, we can see pointed spades. This is done so as to connect the cord element to fit a corner.

3) The interlacing of the cords is consistent; they alternate between "overs" and "unders", which ultimately contributes to an overall spiralling look.

4) The basic pattern repeats.

5) The path of celtic knots is continuous. To explain, this means that if we cut a single cord, pull the two ends, and the panel tightens, then it is a Celtic knot. If it unravels then it is interlace.

Drawing celtic knots : 1a) Iain Bain's 3-cord design

Step 1 Step 2

Step 1 Step 2 Step 1:

Draw a 20cm by 3cm rectangle. (Only a suggested dimension). Draw lines(horizontal and vertical) in between so that the intervals between the lines are 0.5cm. Put small dots at the intersection of dark horizontal and light vertical lines, like what is shown in the figure below. (To check, there should be 2 rows of dots with a height of 1cm.) Note: Do all these very lightly with a pencil.

Connect the dots diagonally(45degrees), forming crosses between the 2 horizontal lines.

Step 3 Step 4

Step 3 Step 4 Step 3:

Extend the set of diagonal lines to form points above and below the initial lines. If you number the top row of dots 1, 2, 3, 4, and so on, from left to right, your peaks should be between 1 and 2, 3 and 4, etc. Not between 2 and 3, 4 and 5, etc. (To check, you will get a zig-zag path-like figure.)

Step 4:

Connect the points with arches. The points that move into the corners are shaped with points. Make sure that the arches are of constant width.

Step 5:

Erase the dots.

Step 6:

Erase the cord lines from each intersection so that one cord goes over the other. Remember that you must alternate between "overs" and "unders" to create a weaving effect. Step 7:

Clean up any stray marks. This stage is known as the knotwork plait.

Break the plait at every 5cm horizontally by erasing the lines where they cross so that we get 3 rectangles each measuring 5cm by 3cm. Then reattach the lines so that the corners are shaped with points, like what we did in step 4. (To check, see that each cord goes back on itself instead of continuing along the plait.)

1b) Iain Bain's 4-cord design

Step 1

Step 2

Step 1

Step 2 Step 1:

Draw a 18cm by 4cm rectangle. (Only a suggested dimension). Draw lines(horizontal and vertical) in between so that the intervals between the lines are 0.5cm. We will label the dark horizontal lines as 1cm, 2 cm, and so on for easy reference; the same goes for the vertical lines. Put small dots at the intersection of even-numbered horizontal lines(2cm and 4cm) and odd-numbered vertical lines. Then put 2 dots on the line: one 3/4 down the square and the other 1/4 down. (To check, there should be 2 rows of dots with a height of 2cm.) Note: Do all these very lightly with a pencil

### Steps 2 to 7 are exactly the same as the above.

Note: Instead of 1 pointed spade each at the top left and right corners only , there should be 1 pointed spade each at the bottom left and right corners as well.

Step 8: Break the plait at every 6cm horizontally by erasing the lines where they cross so that we get 2 rectangles each measuring 6cm by 2cm. Then reattach the lines so that the corners are shaped with points, like what we did in step 4. (To check, see that each cord goes back on itself instead of continuing along the plait.)

2) Marc Wallace's simplified construction of a knot

Step 1:

Draw a grid with even number of squares in both directions. For example, you can draw a 6cm by 6cm grid with 1cm intervals. Put dots at the midpoints of the sides of each small square, where the sides are 2cm.

A splitting is a line through which the knot is not allowed to pass. These lines are drawn slightly shorter than 2cm, and are centered on the dots drawn in the previous step. Place splittings on all the dots along the border of the grid(external spittings).

Step 3:

All internal splittings are either horizontal or vertical, and only 1 splitting can be placed at each dot. Draw the internal splitting as shown:

PS: This is only one example of an internal splitting.

Connect the remaining dots diagonally. However, do not connect them completely from dot to dot so that we have a small distance between the connecting lines and dots. Step5:

Going diagonally outwards from each dot, if the dot is approaching a splitting, draw a short curve out from just past the dot to the edge of the grid square, making sure not touching it. The curve should go straight towards the dot, and should be parallel(at the peak of the curve) to the splitting where it meets the grid edge. Take note that the curve should not touch the dots. Wherever a horizontal splitting and a vertical splitting meet in a corner, the knot will also have a sharp corner. Draw these corners in a way such that they join the short curves smoothly. Step 7:

Notice that there are disjoints(ignoring the spaces between dots) in the figure. Fill in the disjoints by drawing straight-line segments. ## Pencil Drawing Beginners Guide

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### Responses

• Lena
How to draw a celtic knot step by step?
8 years ago