Pythagorean Triangle 3 : 4 : 5
W Bro Vic Brown PGStB
Many thanks to WBro Iain T PGStdB who forwarded the below for use in Devotion News. Iain is the L.E.O. at Baxter Lodge No 934, MAP presenter, and also member of the Kring Nieuw Holland Study Group.
Any triangle whose sides are in the ratio 3:4:5 is a right angle triangle. This has many occurrences in Freemasonry, but most obviously, in the Square.
After a Master installs his successor, a Senior Mason presents him with a Past Master’s jewel, which he is honoured to receive and wears with considerable pride. On the ribbon there is usually his Lodge logo and/or an engraved gilt bar, whilst below is a gilt square with a dangling bit. No one ever explains the significance of these to the proud owner. (Unless you hear the full charge given on its presentation, Ed).
The square can be one of two types; a symmetrical square similar to that on the Master’s Collar Jewel, or an asymmetrical square. It is the latter, which is of particular interest. Each leg is graduated, with one leg 3 units long, and the other is 4 units long, with the end scalloped. Each leg is 1 unit wide. Without the scallop, the distance between the insides of the ends would be 5 units.
This, of course, illustrates Pythagoras’s theorem; that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It is by no means certain that Pythagoras did come up with this proposition, but the Roman Architect Vitruvius, in his book, did credit Pythagoras for the idea. Vitruvius certainly made great use of the proposition in laying out and erecting many of his buildings.
However, it was not until 200 years later that Euclid finally proved Pythagoras’s Theorem with his 47th Proposition. It is the diagram of Euclid’s 47th Proposition which dangles from the PM jewel. A copy of Euclid’s Proposition is appended to this paper. 3 : 4 : 5 is the simplest form of the proposition, enclosing the largest relative area. The next combination of whole numbers is 5:12:13 and I have noticed that some Worshipful Masters on the Continent carry a large silver square on their Collars with that proportion – why, I have not been able to find out. The combinations of whole numbers beyond that all conform to the rule of the shortest side being an odd number and the other 2 having a difference of just 1.
It should be noted that whilst Vitruvius and Euclid gave the construction of the right angled triangle, they did not describe the remaining secrets of the 3 : 4 : 5 triangle known only to a Grand Master Mason. These will be illustrated later. So how does the 3 : 4 : 5 apply to us in Lodge?
On his Initiation, (in some Jurisdictions, not under UGLV) the Candidate is conducted to the East in readiness for his Obligation. He is instructed, “Left foot up the Lodge and right foot across the Lodge”. Note that originally this was not just heel to heel, but right heel directly behind the left heel, thus forming the 3 & 4 units of the triangle. “Take a short pace with your left foot, bringing the heels together, another a little longer and another longer still”. What is the significance of these increasing steps? I think some of you will already be ahead of me! Yes, it is the 3 : 4 : 5. By way of confirmation, in some Scottish Lodges, the Candidate is told to take a pace of about 9”, another of about 12” and another of about 15”, giving the ratio 3 : 4 : 5. In other Scottish Lodges he is told to take paces of 9”, 16” and 25” - 3 squared, 4 squared and 5 squared. So why do we start with the left foot? Formed in a square like this, the left foot is firmly in front and is the natural one to lead off.
The oldest known icon or symbol in the world is the Swastika, being over 4000 yrs old. It is a combination of two Sanscrit words meaning, ‘Good Luck’ or ‘So be it’. The symbol is formed by four Past Master’s squares as below. In Northern Europe it is called the ‘Fylfot’ meaning ‘many-footed’ and representing Thor’s Hammer.
In Greece it is the Gammadion, each arm being formed in the shape of letter ‘gamma’.
India, it is known as the Jaina Cross.
China, it is the ‘Wan’.
The German word is actually, ‘Haakon-kreuz’ or ‘hooked cross’.
Ireland, it is the Cross of St Bridget.
In the Isle of Man, it has just three arms (or legs!)
In Heraldry it is the Croix Pattée.
So what is the connection with Masonry?
To our ancient brethren it represented the heavens and specifically the Great Bear Constellation revolving around the Pole Star, the gap in the middle, which was a reference point for orienting buildings. Many of the old operative masons used the swastika as their personal Mark. The Orientals considered it to be the wheel of sun, good luck and life. There is a special word ‘Deasil’, which means with the rotation of the sun. Interestingly, Buddhists & Hindus always circumambulate their Holy Shrines with the sun.
In contrast, the reverse swastika is considered to be the wheel of the moon, misfortune and death. There is a special word ‘Widdershins’ meaning against the sun. Against advice some 90 years ago, Hitler apparently insisted on using this version of the swastika. He also slewed it by 45°, which is wrong, because the true swastika is always oriented NESW.
Maybe after another 100 years, Hitler will be a vague memory and the swastika will be once again viewed as a sign of good luck.
THE 3 : 4 : 5 TRIANGLE
Let us now see where the 3 : 4 :5 triangle appears in our Craft Masonry. 2 triangles, end-to-end, give the proportion of the flaps on Craft, Mark and Royal Arch aprons.
Side-by-side gives the correct indentations around the Floor cloth – (demo) also the original indentations around the border of the Royal Arch apron, now slightly squashed up.
Four of the triangles together form a special diamond, as on the original Royal Arch sash, now also squashed up a bit. Interestingly, this diamond forms the proportions of certain building façades as exemplified in Appendix B.
This shows the façade of Westminster Hall and how the whole of the front end conforms exactly to this specific diamond pattern, including the pitch of the roof. Similar proportions are to found on York Minster, Beverly and Ripon Cathedrals and also Magdelen College at Oxford.
We now come to the interesting bit and will see how our 3 : 4 : 5 triangle was used to lay out buildings on the 5 point system.
What is the orientation of Cathedrals & Churches? Yes, normally East / West, but not always. Some churches, dedicated to a particular Saint, are oriented to where the sun rises on that Patron Saint’s Day. However, we will now work the normal East / West.
How do we establish the true East / West line? From the sunrise? The sun rises somewhere in the East, but precisely where depends upon the latitude and time of the year – so we cannot use that. However, there is a constant, which is the Pole Star at Low XII in the North, or the sun at High XII in the South. We will use High noon, or to quote our ritual, ‘when the sun is at its meridian’.
Q. Bro J.W. What is your position?
JW In the North.
Q. Why so placed?
JW To see the sun at XII noon, and to inform you when the High Time has arrived.
Q. Perform that duty.
JW It is High XII and the sun shines through the space in the South
Q. Strike a line. (A cord is laid from the JW in the North to the South, lining up with the ‘sun shining’ through a narrow slot).
The next question is ‘where is the starting point for laying out a building’? Is it the NE corner or the centre? Ans. The 1st foundation stone is laid at the NE corner, but the building is actually laid out from the centre.
Q. Grand Master Mason King Solomon, will you fix the centre?
KS. I will.
Q. What is a centre?
KS. That point within a square or oblong building from which each corner is equally
Q. Why do we work with the centre?
KS. Because by that method so long as you work true to the centre the corners cannot err.
Then I mark the centre. (Does so).
There is the centre, work ye to it.
Q. King Solomon, Hiram of Tyre & HAB, now use your rods to establish the true East / West line. (does so & anchors the ends)
See illustration below.
Only the 3 Grand Master Masons had possession of the 5, 4 and 3 unit rods, which were the secret for laying out buildings. Thus all three had to be present to form the right angle. ‘We three do meet and agree’.
We will now use the 5 point system to lay out a square building, such as a pyramid. (The rods are moved 45° and diagonal cords are laid out). The Grand Master Mason states the required equal distance from the centre to each corner, and these are marked. The four foundation lines are laid over the corner marks. These are called the ‘Land lines’. They are anchored beyond the corners, to act as sights, and are called ‘landmarks’ – the substantive origin of the word.
This square layout is geometrically straightforward, but now we will see one of the secrets of the 3 : 4 : 5 triangle. (The land-lines are removed and the rods are rotated to the next largest angle. The diagonals are again laid alongside the rods) The four land-lines are laid again equi-distant from the centre along the new diagonals – and we now have a floor plan with a ratio of length to breadth of exactly 2 to 1. See illustration below.
Finally we turn the rods to the smallest angle of the triangle, lay the diagonals along them, and then the land-lines on the 5 point system.
Lo and behold, we have the length to breadth ratio of exactly 3 to 1, generated entirely by the 3 : 4 : 5 triangle. So what was the size of King Solomon’s Temple? 60 cubits by 20! (3:1)
This ‘secret’ property of the 3 : 4 : 5 triangle was apparently unknown to Pythagoras and Euclid. Having laid out the plan, what happened before the 1st footing stone was laid? A human sacrifice was made. In those days it was considered an honour and they actually had volunteers! (Nowadays they are known as suicide bombers). 1st footing cornerstone is always laid in the N E. Why? In order to catch the first light of day at 6 am with the remainder following the sun, i.e. S E at 10 am, S W at 2pm and N W at 6 pm. Each corner had its individual human sacrifice. Later on, animals were sacrificed, and nowadays of course, time capsules are buried for posterity. The four corner stones were thus laid first and fully checked out. Only then were the remainder of the foundation stones laid in between. Incidentally, the emblems on a MM apron are usually but inaccurately described as ‘Levels’, despite not looking anything like a proper level as depicted on the SW’s pedestal. No, these emblems are the cross-section of the huge footing stones to take the massive weight of the finished building. Concrete had not been invented in those days. The corner footing stones were carved with this cross-section and it has been calculated that for a building the size of King Solomon’s Temple, each would have weighed 45 tons.
By now, you will have appreciated why in Lodge we follow the ancient custom of circumambulating the Lodge with the sun.
Subsequent courses were laid, one complete course at a time. Ell-shaped stones were used at the corners, alternating short ells and long ells. All of these stones were carved and aligned by hand with an accuracy which nowadays really beggars belief, especially realising the very heavy weights involved. Those who have taken the Nile trip from Luxor to Aswan, and seen all these temples, still standing after more than 3,500 years, without the use of any cement, will share my wonderment at the skill and knowledge of those ancient masons. They must have used certain techniques that have been lost in the passage of time.
Finally, the name of God used by our operative predecessors is ‘El Shadai’, as recognised by many Royal Arch Companions. You may be aware that all Hebrew characters have a dual function as both letters and numerals. The Hebrew characters for ‘El Shadai’ have the numerical value of 345. Is this just coincidence?
For some reason, of which I am unaware, our ancient brethren measured their pillars and columns by circumference, rather than by diameter. “The height of those pillars was 171⁄2 cubits, their circumference 12 and their diameter 4”. Similarly, the numerical value of the Hebrew characters for ‘Shadai’ by itself is 314 – the ratio of circumference to diameter or π. Is this just coincidence again?
Incidentally, King Solomon’s Temple had its main entrance in the East, with the Sanctum Sanctorum in the West, which was the norm for places of worship at that time. When the Christian Cathedrals came to be built however, the main entrance was located in the West. In like manner, the great majority of Christian Churches have their entrances in the SW.
Brethren, that completes my talk on 3 : 4 : 5 and I hope you will now look at the Past Master’s jewel in a different light, knowing that it represents much more than just ornamentation.
EUCLID’S PROPOSITION 47.
THEOREM In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle.
Let ABC be a right-angled triangle, having the right angle BAG: the square described on the side BC shall be equal to the squares described on the sides BA, AC.
On BC describe the square BDEC, and on BA, AC describe the squares GB, HC; through A draw AL parallel to BD or CE; and join AD, FC.
Then, because the angle BAC is a right angle, and that the angle BAG is also a right angle, the two straight lines AC, AG, on the opposite sides of AB, make with it at the point A the adjacent angles equal to two right angles; therefore CA is in the same straight line with AG. For the same reason, AB and AH are in the same straight line.
Now the angle DBC is equal to the angle FBA, for each of them is a right angle. Add to each the angle ABC. Therefore the whole angle DBA is equal to the whole angle FBC. And because the two sides AB, BD are equal to the two sides FB, BC, each to each; and the angle DBA is equal to the angle FBC; therefore the triangle ABD is equal to the triangle FBC.
Now the parallelogram BL is double of the triangle ABD, because they are on the same base BD, and between the same parallels BD, AL. And the square GB is double of the triangle FBC, because they are on the same base FB, and between the same parallels FB, GC. But the doubles of equals are equal to one another. Therefore the parallelogram BL is equal to the square GB.
In the same manner, by joining AE, BK, it can be shown, that the parallelogram CL is equal to the square CH. Therefore the whole square BDEC is equal to the two squares GB, HC.
And the square BDEC is described on BC, and the squares GB, HC on BA, AC. Therefore the square described on the side BC is equal to the squares described on the sides BA, AC. Wherefore, in any right-angled triangle &c. Q.E.D.
The 3, 4, 5 Triangle and the Diamond in Architecture
Similar ‘diamond’ proportions at York Minster, Beverley, Ripon & Magdalen College, Oxford