MINOTAUR MEETS MAPMAKER
Maths & Maps – Class 7 – Middle Lesson
“So you’ve found your way into my Labyrinth? Finding the way in has never been a problem;” said the Minotaur in menacing tones to young Theseus, the Class 7 teacher “it’s finding your way out that’s the tricky bit!” The Minotaur was indeed awesome, terrifying one might say, but as Theseus entered his great cave with purpose, he was inwardly braced for this grand spiritual spectacle before him.
Theseus was on a quest for inspiration to enliven his Maths in Maps middle lesson with his 13-year-olds. Knowing that mathematics was (of the 12 zodiacal subjects) that ascribed to Taurus, this was where his spiritual seeking had led him.
With Taurus’ he didn’t know quite what to expect – the great man-headed bull of Babylon? The winged bull of the Gospel of St. Luke? Even the bull of Mithras, with its ancient Persian origins! It was just his luck to have the most frightening of all, the Minotaur of the Greeks; which was how they perceived the eternal Taurus reality.
Taurus was the sign which preceded their own Aries; the Greco-Roman cultural civilization enjoying the Ram rising in the vernal equinox for their 2160-year cultural dispensation. Taurus was the sign of Egypt-Chaldea-Babylon-Assyria it al! The Bull was actually seen as inimical to the evolutionary aspirations of the Greeks – a spiritual retardant even.
As such, it was depicted in uncomplimentary images, devouring virgins and the like. Taurus has ascribed to it the Sense of Thought, a necessary faculty for entry into the Labyrinth of maths. This sense of course is brain-based, the Labyrinth being a picture of the cerebral convolutions, and in an esoteric sense, the maze of concepts itself. Mathematical thought is the most complex thought of all.
“So before you ‘slay’ me – ha, ha, ha – which in the occult sense thankfully means only to ‘master’ me, you have to convince me that, as an educator, you are worthy of receiving my profound Number Inspiration.” Growled The Minotaur threateningly “Oh I know that you’ve sensibly taken the precaution to lay down the ‘string’ so you can find your way out of here. The string image was one of Rational Soul linear intellect; today teachers need not mere intellect, an intrinsically heartless quality, but intelligence, a higher kind of thought.
“With the advent of the 3rd millennium.” The Minotaur continued amiably “the New Age of Abraham, that intelligence must be spirit-imbued. So tell me about the Maths in Maps middle lesson.”
“Well, er, you mean there’s a chance I won’t be able to find my way out of here!!?”
“Why are you so special? There are legions of intellect-hardened ‘seekers’ wandering around in my endless corridors right now – so get on with it!”
“cough – the, um, middle lesson balances out this tendency to intellectualism – an inherent abstraction – by its maths emphasizing the artistic, experiential, heart-based aspects. Maps of course give limitless opportunity to present the human face of number. Numeracy, the 2nd of the 4 middle lesson streams, is, as you know, the Taurus, er, Minotaur! Subject – that’s why I’m here.
Of the 3 numeracy strands, expressing will, feeling and thinking as they do, Maths in Maps is that of ‘feeling’ – it is Diagrammatic Number. There are few areas in maths where the opportunity to represent number as picture is so apt. So the psychic profile of this unit is…heart/middle – sense of thought/numeracy – feeling/diagrammatic!”
“Okay, so far so safe; that certainly provides a basis of higher reality for your teaching – so tell me of the type of content you intend your 13-year-olds to enjoy for the next 3 weeks.”
“Um, I thought you might be able to actually help me on this?”
“I’ll help when you need help! Now get on with it – why teach the subject in Class 7 anyway?”
“Well ‘finding your way’, which is the kernel of map – and Minotaur – consciousness, is intrinsically related; not just to the earth, being a kind of geography as it is, but to the heavens as well. So much orientation (‘to face east’) is based on the celestial map.
Class 7 children are standing on the threshold of their new 7-year astral/soul or ‘star’ development; as such they are introduced to Astronomy this year – science of the stars. To introduce an intensive star-based map awakening earlier would be pedagogic force-feeding.”
“Hmmm, that seems a sound basis for this particular timetabling – what sorts of maps are you going to teach them about?”
“Oh all kinds really; in fact I’m going to collect interesting maps from all over to create a living cartographical center in the classroom. This way I’ll be able to make instant reference to one feature or another as I’m teaching. There’ll be the usual world maps; but as well I’ll have globes (both terrestrial and celestial); antiques maps; council; star; moon; ocean; topographical; navigational; atlas – and more.”
“In fact all these different kinds of maps will help me demonstrate – in a transformed way of course – how all maps fall into four main groups; based on body consciousness. Globes, the most visually accurate of all cartographic representations, are a Physical Body expression; the sphere is the timeless symbol, and realityof this Old Saturn creation.
General land maps, in all their variety from pirate treasure maps (we’ll make one of these!) to your standard map of Australia, depict, in their form of the earth, the Etheric Body – body of ‘formative forces’. Because of this supersensible creation factor – the Elohim, Spirits of Form created the earth – we never really tire of looking at maps. Here we see in a simplified form the world of those ether beings, the geomantic dragons of mountain and river.
The Astral cartographical expression is the chart (or ‘cart’); that used so skillfully by mariners old and modern. Sea persons never refer to their maps, but always to their precious ‘charts’ – precious? Without them they are as good as blind at sea. A navigational chart, say of the Central Pacific, may contain no topographical features at all; but astral or star-based information, like position and direction, a-plenty.
The ocean, or water in general, is the astral element, good navigators of old were astral initiates of one kind or another; with adventures like Cook and Flinders belonging to a very exclusive club, the Maritime Lodge, an original Henry the Navigator creation!!
The 4th type of map, the plat, is more Ego-informed, being one of ownership based on property boundaries. These are used by councils and legion other government agencies. They deal not so much with the earth reality (most don’t have physical features at all) but with proprietary rights. The ego of the community imposes its own reality on the landscape, usually in the form of the square.
Plats become a patchwork of squares and rectangles so well understood by mining companies, developers, farmers and other land users. These of course are little appreciated by children, whose more healthy instincts are drawn to globes – the word plat means flat!”
“Hmmm, I look forward to chatting about these things with you – for millennia! Said The Minotaur, his tone agreeable “But it strikes me that in all this fascinating content, you could lose sight of this as a maths lesson. I do have to defend my own plat areas you know; after all, cartography is strictly speaking a part of the geography stream.”
“Correct! In fact there is a major Cartography middle lesson in class 9, two years later; this is supported b y a practical afternoon block lesson on Orienteering. Yes, this map description segment is relatively brief – and where possible accompanied by number interpretation.”
“Good, although the less number-loving teacher could succumb to the temptation to make it merely an interesting maps lesson.” The Minotaur added suspiciously “What else?”
“Well there’s the history of mapping – er, with ample number descriptions of course! Most indigenous peoples didn’t need maps at all, having an intimate and knowledgeable relationship with their heavens and earth. When Aborigines traveled they never got lost – they followed the ‘song lines’ of the earth…rather than grid plotting!
When civilization imposed its own imperatives, with the building of towns, canals, harbors, etc., the need to document these became ever more urgent. The birth of mapping coincided with a separation of man from his/her environment; requiring an abstract version of it – a map.
The first known came from your won – hmmm, odd? – period. Babylonian clay tablet maps have been found dating from around 2000BC. Naturally the ever-clever Greeks, whilst renouncing other Taurean aspects, found map-making germane to their culture; with roman Ptolemy drawing a ‘world’ map which even contained lines of latitude and longitude. And guess where he positioned 0° longitude?!”
“Right where it is today, at Greenwich of course.”
“How did you know that?!”
“Anyway, with the expansionist policies of Rome, maps began to take on the present 2-fold function as statements of geographical and political reality. But it was the Arabs, up to about 1000AD, who made maps really accurate, with their sexagesimal (60°) grid system. Arabs were an ocean-going culture, or many of them at least; so different from the land-lubbing Babylonians and Greeks. So navigational demands provided great advances in mapping.
The invention of the printing press some 500 years later brought maps within reach of the common person – prior to that these hand-drawn masterpieces were very precious items indeed. The first map was printed in Bologna in 1477. The first globe, indicating a major planetary awakening for European civilization, appeared in 1492. This was the same year that the intrepid Columbus set sail for the New World. – or Asia as he thought. He ventured forth under the heretical idea (for Europeans anyway, most other cultures weren’t so ignorant) – that the world was a free-floating sphere.
With the – again expansionist – adventures of the maritime European nations, especially Holland, map science preceded apace, with the proliferation of various ‘projections’, like Mercator and Mollweide – plenty of number work here!”.
I wondered if I had The Minotaur’s attention as I continued, as he appeared to be dozing “The Dutch and other map creators faced the daunting task of transposing the skin of a globe onto a flat sheet of paper. This of course is impossible, but the various compromises serve specific functions. The Mercator projection allowed mariners to plot courses in a straight line – in spite of the absurd enlargement of land masses towards the higher latitudes.
Whipping ahead, brief descriptions of modern map-making techniques were given, especially with the new technology of satellites and aerial photography. By the way, we drew those 3 main projections, the cylindrical, conic and azimuth. This was a very good geometric exercise, giving elasticity to thinking.”
“Ah, numbers at last.: exclaimed The Minotaur, raising his great head as if to sniff the air “I suppose that exercise provided the opportunity to describe the world grid, based as it is on circles of latitude and meridians of longitude. You realize of course that a line of latitude, like Puck and his ’40 minutes’, girds the whole globe; while that of longitude is only a semi-circle? 0° longitude, Greenwich, only goes from the North to the South Poles – then it becomes 180° on the other side of the world, The International Date Line in the Central Pacific; these are the ‘great’ and ‘small’ circles – I love numbers!”
“Yeees, I know – there are lots of number concepts in the magnetic compass too; as well as describing the difference between grid and magnetic north (a 7° east variable at Bryon Bay), I will get the class to draw an accurate – and colorful – compass rose.
This will contain all 16 nominal directions, moving clockwise these would be: north; nor-nor-east; north-east; east-nor-east; east – and so forth. They’ll also design a ‘cartouche’, or title block – again artistically as of old. Did you notice that as maps became more accurate, more based on hard scientific data, they became less pictorial?
Well at the stage my children are in – recapitulating the Age of Chivalry – I intend to retain the inherent aesthetics of map-making, with decorative dragons, wind cherubs, spouting whales, old shops and other embellishments. Anyway, the cartouche gives me opportunity to describe and explain all the vital information it contains, like color keys, publishing info., and – numbers again – scale!
Scale is divine mathematics, allowing us to realize the miracle of transforming a whole planet into its microcosm, in the case of a globe, fairly accurately. We do scale exercises, like 1:1,000,000, 1:500 and so on; this allows us to be cartographers ourselves. We could map the classroom; or the school grounds – or each child their own home.”
“These maps could even include keys, elevations and topographical information, and grid plotting.” I concluded.
“Speaking of position-finding,” The minotaur added “there are plenty of atlas exercises (every child should have their own atlas) where they are given a co-ordinate and asked to find, anywhere in the world, what town, mountain, etc. is to be found there. This works the other way as well, where they’re give a town, and asked to find the co-ordinates. Children love atlas work – it’s a kind of vicarious globe-trotting.
Finding north with a digital watch could add light relief for the desk-bound; merely (in the Southern Hemisphere) hold the watch over your head, so you can see the face, and point the hour hand to the sun – any time of day, North is the point exactly half way between the hour hand and the 12. Who needs a compass?!”
“What about at night? – ah, I know, you find north by default, by finding south! Count down 4 ½ times the length of the Southern Cross, and run an imaginary vertical line to earth – that’s south!”
“What if you can’t see the Cross?” said a puzzled Minotaur.
“What if you can’t see the sun?!!” I wished I hadn’t been so impatient, as my bovine companion scythed his huge horns in a most alarming way! “Sorry – er, I know you’re just trying to help – you don’t have to get ‘cross’ – ha, ha…perhaps if I describe the Create Circle I mentioned earlier, a prelude to spherical geometry in later high school!
Here I can demonstrate how it’s shorter to get to Chili from Tasmania by flying south over Antarctica than east across the Pacific. And while on circles and horizons, I might introduce – maths again! – the formula D (distance) = 89.433 square root h. This finds the distance to the horizon from say a ship’s crow’s nest, or a mountain; ‘h’ is the height of observer from the ground. Measurements must be in miles.
A great Circle is the globe cut through the middle; any circle of longitude is a great circle, but only the Equator of the latitudes is one. All other latitude lines are Small Circles, like cutting an orange not through the middle.
This leads on to Eratosthenes’ amazing feat of determining the length of the Equator (24,902 miles). He knew that there was a sun shadow variation of 7 ½ ° over 500 miles, from Alexandria to Athens. As a circle, the world, has 360°, he knew that 7 ½ ° was one forty-eighth of the globe (in this case longitudinally) – merely by multiplying 500 by 48, he arrived at 24,000 – near enough!
So here’s a joke: a hunter – make that wildlife photographer! – traveled south from his camp for 10 kilometers, then east for 10ks, ‘shot’ a bear, then journeyed back due north 10ks to his camp. What color was the bear?”
“…?!…what’s the question again?”
“Ha, ha, ha, can’t get it eh?! So much for the fable Cave of Convolutions! The bear was white – it was a polar bear!! You see, with these directions and distances, the wildlife photographer’s main camp must have been at the North Pole; the only direction you can travel from there is south, ha, ha…what’s wrong?! I thought you’d like number jokes, where’s you sense of humor?”
“We divinities don’t suffer tricksters gladly – you’re in The Labyrinth to stay schoolteacher!”
“Wo! I’m outta here – now which way? – ah, lucky I trailed that string behind me when I came in – now where’s the end…?!”
A really good compass plotting exercise for students is to select a goal, such as a towel on a wide stretch of beach (so they don’t run into anything when ‘blind’); plot a course from a point some distance away, such as, “12 paces north-east, the 20 paces due west, then 4 paces nor-nor-west – and so on. Then, one-by-one place a thick towel over their heads, spin them round to disorientate them., give them the course map, and under the towel they have to walk the course using only a compass. It both surprises and amazes them how close they can get to the goal without seeing it, the compass alone being their eyes.
Another excellent plotting exercise is to do with locating something under water. Sink a marker, such as a brick, in water shallow enough to duck dive to the bottom, but deep enough to conceal the brick. Then on the surface directly above the brick take two visual marks, lining up first, say a chimney with a tree some distance away, and then two other alignments in another direction (radio tower and rock outcrop?). Return to shore. On returning later (maybe in a week or so to increase the “Wow!’ factor), swim out to the same general area, line up the chimney and tree and the other two alignments exactly as before, and dive down. It’s amazing to find the brick directly below from just two alignments of each of two objects, maybe a mile or so away! This is even more accurate, with a little practice, than satellite positioning technology.