Geometria
One plus two plus three plus four equals ten, proving the first four numbers are the sum of number. x squared equals x whether x equals one or x equals zero, proving one does not equal zero. Any number squared divided by two plus half the original number is equal to the number of times each of the consecutive integers in the original number combine with itself and each of the others once, as in dominoes. Pi squared and the square root of pi are both constant. A place holder exists that is neither one nor zero. What is the place holder?
When we use equals we are saying some entity is the same as a different entity. When we use does not equal we are saying some entity is not the same as a different entity. By universal abstraction we can define this as to say being is being and the contradictory being is not being. That then explains the way in which we use equals, which is the mathematical equivalent of the word "is". And the way in which we use does not equal, which is the mathematical equivalent of the word phrase "is not".
Given that understanding, it becomes clear that if we attend to the matter "what is equals?", we will be forced to attend to the more abstract matter, what does "is" mean? The important knowledge here is to realise that we can not answer the second level question "what is equals?" unless we can answer the first level question "what does <is> mean?". And if we can answer the first level question, then answering the second level question becomes easy.
The concepts that are inter-relationally involved in this kind of study include being, doing, meaning, existence, non-existence, identity, non-identity and no other concepts. In mathematical terms we can say the concepts are one, some, more, less and none. Therefore, if we consider only those five mathematical concepts we can define some basic terms. One object is an existant identity. If the same identity is repeated that is some. Some is more than one. One is less than some. None is the non-existance of one and some. The concept of more is to do with addition. The concept of less is to do with subtraction. A repetitive addition is called multiplication. A repetitive subtraction is called division. Existence is an affirmation. Non-existance is a negation.
Any number squared divided by two plus half the original number is equal to the number of times each of the consecutive integers in the original number combine with itself and each of the others once, as in dominoes. In a standard nine bar set of dominoes the integers one to nine and zero are all represented in relation to each of the other integers. That is a total of ten units. Ten squared is one hundred. One hundred divided by two is fifty. Half the original number is five. Fifty plus five is fifty five. And in a standard nine bar set of dominoes there are fifty five counters. Given dominoes do not use late language sign label for number, and instead use a pattern of points to show the universal abstract that exists at the earlier level than sign label, they are useful in regard to demonstrating the ontological status of existant being of number.
At the level of dominoes, the sign labels and the concepts do not yet exist. Without the sign labels and without the concepts, still dominoes exist. Dominoes therefore are ontologically earlier than the sign labels and the concepts. And from the ontological existence of dominoes it is very easy to build the sign labels and concepts of number.
If we imagine we have a set of dominoes for only the units one and none, then that is two units. Two squared is four, four divided by two is two. Half the original number is one. Two plus one is three. Therefore we will have a set of dominoes with three counters. One of those counters will show none on both sides of the counter. Another will show one on both sides of the counter. Another will show none on one side and one on the other side of the counter. In that way without defining our concepts we are showing the concepts in their immanent form. And with the dominoe that has none on both sides we are saying none is the same as none or none equals none. And with the dominoe that has one on both sides we are saying one is the same as one or one equals one. And with the dominoe that has none on one side and one on the other side we are saying none is not the same as one or none does not equal one. And we know what we mean because we have three dominoes and we can see what we mean. And we think that each of the three dominoes is different to the other two. And if we thought of the connecting line between the two sides of any one dominoe as plus, then we would know that none plus none is none, one plus none is one, and one plus one is two. And if we look at the whole dominoe and thought of what happens when we take away one side that would be like minus, so we could say none minus none is none, one minus none is one, one minus one is none, two minus one is one.
All we need now do is notice what are the functions that are evident without specific sign label. And we can see that the difference between nothing and something is similar to the difference between the dominoe that is blank on both sides and the dominoe that has a blank on one side and a one on the other. And the difference between one and some is similar to the the difference between the dominoe that has a blank on one side and one on the other and the dominoe that has a one on both sides. And we can understand that some is more than one and one is less than some and none is the non-existance of one or some.
So at this stage we have proved the first five concepts of number which are one, some, more, less, none. And we can apply those concepts as sign labels of functions such that we understand one is existant entity, some is a repetition of one, and none is the non-existance of entity. And we can apply the sign label functions of addition to some being more than one, and subtraction to one being less than some. Which means we have now invented our sign labels for the functions of addition and subtraction.
And we know what we mean by "is" because "none <is> none" and "one <is> one" and we know what we mean by "is not" because "none <is not> one". Therefore we have demonstrated using three dominoes the function before sign label of the concepts of one, some, more, less, none, addition, subtraction. And now we simply invent the sign labels to specify those functions.
For more information as to dominoes and game theory please follow the hyperlink: http://en.wikiversity.org/wiki/Game_domain_model And see also:http://en.wikiversity.org/wiki/Zero_unity_and_infinity
Given the previous discourse then we can say that one is the equivalent of a point. And we can provide an initial definition of a point. That is: "A point is defined as the prime difference between none and some". Or alternatively we can say: "A point <is not> none and also a point <is not> some". We can say that a point is more than none and less than some. A point is that which exists if we do not have some and do not have none. The first difference between nothing and something is a point. Therefore in all these propositions a point is given in terms of "difference", which leads us to say that a point is the prime difference from none or nothing. And a point is the first condition of some or something.
In this sense we can negate a point by saying "none exists" or "some exists". Both none and some are different to a point. And a point is the absence of none or some. A point exists that is less than some and more than none.
Geometry in this system is defined as the study of the relationship between triangle, square and circle and nothing else. Triangle, square and circle are all examples of "some". Therefore, before the study of the relationship between triangle, square and circle we must understand the earlier matter which is a point.
The fastest possible construction of three dimensions is point, line, plane, tetrahedron. Tetrahedron and the other four prime solids are all based on the study of the relationship between triangle, square and circle. Therefore the geometric system being studied is only based on the following figure:
1. Make a point.
2. Extend a line from the point.
3. Bisect the line through its centre.
4. Draw a third line equidistant from the point of the bisection such that the third line crosses each of the extensions of the first two lines and both ends of the third line are connected together.
5. Draw four connecting lines from the point where the first two lines touch the circumference of the circle.
That figure is the circle, square and triangle in relationship. It can be alternately described as "that geometric figure made by the circle circumscribing a square divided by four similar triangles".
Rotate the same figure by one quarter turn of ninety degrees around any one of its corners to obtain a mirror reflection. And by another half turn of one hundred eighty degrees around the same corner to obtain the larger square. Circumscribe the larger square by a circle. That gives the larger square containing four exact versions of itself. Any of those four smaller squares could contain four exact versions of themselves inside themselves.
Half the larger square gives a rectangle formed by two versions of the smaller square. Reflect the two extreme triangles at either end of the rectangle outward from the extreme edges of the rectangle. Extend a line from the end point of each extreme triangle through the centre of the entire figure. That gives three horizontals, seven verticals and twelve diagonals meeting at ten points which is a total of twenty two plus ten, which is thirty two.
Therefore we can hypothetically suppose an association between the given system of geometry and the mathematical system of Mozart in terms of three, seven and twelve. Where the three scales of the grand staff are only the repetition of one scale of seven tones, and the seven tones are only the combination of twelve half tones. That is three, seven and twelve. For further study of the mathematics of music please follow the hyperlink: http://en.wikiversity.org/wiki/Patterns
Before our geometric system can be used to study the five socratic solids it is necessary that we obtain the information provided by the study of the geometric figure made by the circle circumscribing a square divided by four similar triangles. Before we can obtain the information provided by that figure we must prove the point, line and plane.
Our definitions are as follows: A point is neither none nor some and is the prime difference between none and some. And a point is the absence of none and some. Or a point is more than none and less than some. A point is the possibility of the existant being that is not none and not some. And we prove a point by our system of dominoes of three counters by saying:
None <is> none.
One <is> one.
One <is not> none.
A line is extension in any direction from a point. A line is the prime difference between one and some. A line is the precondition for some to exist. Given any two points then a line is the shortest distance between the two. A line is the element required in order for a plane to exist. A line is length without breadth. A line is the possibility of the existence of verticality or horizontality. A line is proved in reality by the plumb along a vertical or the spirit level along the horizontal.
A plane is extension of line along a vertical and horizontal axis. Or a plane is that entity made possible by the bisection of one line by another line. Or a plane is that figure made by extending a line equidistant from a point such that both ends are connected. Or a plane is that figure made when any three straight lines have all of their ends touch only one end of each of the other two. Our proof of the plane is the circle circumscribed square and square divided into four similar triangles.
Object is plane surface in two directions. Object is that entity made possible by the inter-section of plane along the horizontal and vertical axis. As plane is to line, so object is to plane. Object is the extension of line along two horizontals and one vertical. Object is the possibility of extension along length, breadth and depth. The first object is that entity whose entire surface is equidistant from its own centre of gravity, or an infinity of lines of all one length extended from a point in every direction, known as a sphere. The prime elements of the first object are tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron. Our proof of object is the inter-section of lines provided by plumbline and spirit level.
The preferred definition of pi in this system is twenty two divided by seven since it again refers to the three, seven, twelve, twenty two structure. And the closer ratio of three hundred fifty five divided by one hundred thirteen. The length of the circumference of a circle that circumscribes a square divided by the diagonal that connects any two opposite corners of the square is the relationship that we must attend to in particular. Since it means that the diagonal between two opposite corners of all and any square when divided into the length of the circumference of the circle that circumscribes the same square gives a constant ratio. That is the ratio known as pi and in rough terms it is always twenty two divided by seven. This particular idea of geometry is only interested in relationship between entity rather than in the entity individually. Half the diagonal line from each opposite corner of the square is the radius of a circle that being the length from the centre of a circle to its circumference, and in a sphere the length from the centre of gravity to its surface.
Pi is equal to three point one four one five nine two six five three five eight nine seven nine. Or 3.14159265358979. Pi is an irrational transcendental number.
The squaring of the circle is only meant to refer to that entity where the perimeter of the square is the same length as the circumference of a circle. It gives a different figure to our basic design such that each of the four corners of the square extends just beyond the circumference of the circle and the circumference of the circle extends just beyond the edges of the square. It is easy to construct with the same level of precision as twenty two divided by seven simply by using a piece of string of any length. Use the piece of string to make the circumference of a circle. Use the same piece of string to make the perimeter of a square. The piece of string does not change length. The circumference of the circle and perimeter of the square will be found to make only the shape described and never any other shape, that is where the corners of the square extend beyond the circumference of the circle and the circumference of the circle extends beyond the edges of the square. So in this figure where circumference of circle and perimeter of square are equal length we do not have the same relationship of diagonal of square to circumference of circle known as pi.
Where in the original given figure the diagonal of square is one large triangle and the large triangle is double two similar small triangles then half the diagonal is one side of one of the smaller triangles. Meaning that in the same figure the side of one of the smaller triangles that is not the edge of the square is also the radius of the circle that circumscribes the square. That is again the relationship that we study in this system of geometry between the circle and the triangle. Since it means that any circle of any size has a radius that is one edge of a triangle that can be used to make only a square where each corner of the square touches the circumference of the circle.
It is in particular the relationship between these three entities where each one necessitates each of the others in a constant permanent relationship that we are interested in. Again, what it means is only the kind of triangle where one angle is ninety degrees and the two sides of that angle are the same length. And only the square that can be made from that one kind of triangle. And only the circle that circumscribes the square such that the diagonal of the square is the diameter of the circle and half the diagonal of the square is one of the sides of the ninety degree angle of the triangle which is also the radius of the circle.
This then is the geometric ideality of the Platonic forms rooted in the Socratic discourse of the Republic and nothing else. It entirely depends on the system described by Socrates in regard to both the forms and the five prime solids. In particular the principle that is attended to is called constance which is close to the virtue known as temperance. Because the kind of triangle in relationship to square and triangle and square in relationship to circle is constant in their relationship this provides an eternal ideality which we restrict our study towards. The content of the Socratic discourses in particular provides the basis for this.
Constance of relationship is not only found in pi. It is clear that in the described figure the length of the side of the square is also the hypotenuse of the triangle. That the length of the side of the square and hypotenuse is in constant relationship to the radius of circle and the other sides of the triangle as well as to the diameter of circle and diagonal of square. And that the length of diagonal of square is also diameter of circle. And that the centre of square and circle are the same. The various ratios of each of these constant relationships is a quantifiable amount that is always the same. So this system of geometry involves both knowledge of what are these constant ratios and also what are the ideas in terms of tangent and sine that inform us as to these ratios. Since the specific tangent and sine quantity will always be the same in this figure, therefore eternals.
The following idea is hypothetical at this point. It is directly related to the triangle, square and circle figure that is the focus of this geometry and is one example of the sort of study that is cultivated in this system.
The peculiar fact that is noticed is this:
a. Given a line of six units in length then the square of that line is six multiplied by six which is thirty six. Whereas half the length of the line is three units in length. So the square of half the line is three multiplied by three which is nine. That means that half the line squared plus half the line squared is nine plus nine which is eighteen. Which seems peculiar. Since the expectation may have been that half the line squared plus half the line squared would equal the whole of the line squared, which it does not. Specifically half the line squared plus half the line squared equals half of the whole line squared. Now, to see if that holds true for a line of a different length.
b. If the whole line is ten units length, then the square of the line is one hundred units, that being ten multiplied by ten. And the length of half the line is five units, then the square of half the line is five multiplied by five which is twenty five. And the square of half the line plus the square of the other half of the line is twenty five, the two combined being fifty. And fifty is half the amount of the square of the whole line which is one hundred.
c. Therefore, given that result we would make the claim that: the square of a line of any length is equal to double the square of half the same line plus the square of half the same line. Or alternatively stated: given a line of any length, then the square of half the length of that line plus the square of half the length of that line is a specific amount that will be found to be exactly half the specific amount that is the square of the whole line.
d. And to state what is not the case is that: given a line of any length then the square of that line does not equal the square of half the line plus the square of half the line.
e. Given that knowledge then we can apply it to any square. And say that one side of any square must be the hypotenuse of a triangle formed when diagonal lines from the two corners of the same side are extended to the centre of the square. Since to extend lines from the two corners of one side of any square such that the two lines meet at the centre of the square is to make a right angle triangle where the side of the square is the hypotenuse. And that means that the side of any square when multiplied by itself must equal the diagonal formed from one of the corners of the square when extended to the centre of the square multiplied by itself and doubled.
f. Since the diagonal formed when extended to the centre is exactly half the diagonal of the line extended from any corner of the same square extended to the opposite corner, it may have been thought that this would result in a contradictory outcome, because it would look as if the hypothesis led to the claim that the side of any square multiplied by itself is equal to the diagonal line between any two corners of the square multiplied by itself. Since the diagonal line between two corners is double the diagonal line that extends to the centre. But it does not lead to that contradiction.
g. The question asked is: Is it the case that the side of any square multiplied by itself does equal half the diagonal line of the same square extended from any corner to its opposite corner when it is multiplied by itself. Since that would seem to be the same situation as that described in point c. above.
h. To clarify: 1.Given a line of six units the square is thirty six. And half the line is three units the square of such being nine, which when doubled is eighteen. And eighteen is exactly half of thirty six. 2. Given one side of a square. And given two diagonal lines drawn from two adjacent corners of the side to the centre of the side. And given that the square of one diagonal is added to the square of the other diagonal. Then the combined amount of the square of the two diagonals does equal the amount of the square of one side. 3. And either one of the diagonals could be extended through the centre of the side to the opposite corner. 4. And that diagonal line that extends from one corner of the side to the opposite corner of the side when squared must equal double the amount given in point 2.
Therefore if the previous hypothesis is correct it suggests that there is a constant relationship between the square of the side of any square and the square of the diagonal between two corners of the same square, in a proportional relationship.
{{{ This note contained within three brackets is a later addition to the original article. If the circumference of the circle is twenty two units, where a unit could be any constant length. And if in rough terms the diameter of the circle is seven units. Then the diagonal of the square would be seven units length. Since that is the hypotenuse of a triangle formed from one half of the square that would mean that one side of the square is five units length, in rough terms. Because, seven squared is forty nine. And five squared is twenty five. So by adding the square of two sides opposite the hypotenuse together we obtain fifty which is only one unit off forty nine. Furthermore if we were to fold one side of the square by ninety degrees to obtain a single line of ten units length then the square of that single line would be one hundred which is twice the square of half the line plus the square of half the line. Given that it means that there is a rough figure of five which must be relative to the seven and twenty two of pi. Because if the circumference is twenty two, the diameter is seven, and if the diameter is seven then the side of the square is five. So twenty two divided by five is four point four. Then a relationship between 4.4 : 3.14 must exist.
Furthermore, if the diameter is seven then the radius is three point five. And that means that the side of one of the four smaller triangles is three point five. Where the hypotenuse of that triangle is five. So three point five squared is twelve point two five. So the square of the hypotenuse of this triangle is twenty five. And the square of the other two sides added together is twenty five. That gives us the root numbers of three and twelve in relationship if we round off to whole numbers. If we keep the points then we have three point five, twelve point two five and twenty five.
At this stage it is clear that we must become more accurate in our figures. Pi is equal to three point one four one five nine two six five three five eight nine seven nine. Or 3.14159265358979.
Using this quantity as a measure of the accuracy that is required, then all we need to know is what are the more accurate quantities for the numbers twenty two, seven, five, forty nine, fifty, four point four, three point five, twelve point two five, and twenty five.
The calculation is as follows:
If the side of a square is five units, where the units are of any length.
Then the square of the side is twenty five units.
And the hypotenuse of the square is the square root of fifty, since fifty is the sum of the square of the other two sides.
The square root of fifty is seven point zero seven one zero six seven eight one one eight six five four seven five. Or 7.071 067 811 865 475.
So the hypotenuse of a square with the side of five units length is seven point zero seven one.
So the diameter of the circle is 7.071 067 811 865 475.
And the radius of the circle is three point five three five five three three nine zero five nine three two seven three eight. Or 3.535 533 905 932 738.
If we multiply the hypotenuse of the square which is also the diameter of the circle by pi then we obtain the circumference of the circle which is twenty two point two one four four one four six nine zero seven nine one eight one. Or 22.2 144 146 907 9181
So now we can suppose a square with the side of five units. The diagonal of seven point zero seven units. The radius of a circle of three point five three units. The circumference of a circle of twenty two point two units.
Where 22.2 divided by 7.07 is pi calculated as 3.1400. And 22.2 divided by 5 is 4.44.
So we can suppose a constant of 22/7 = 3.14 and 22/5 = 4.4 therefore 3.14:4.44 as constant.
Out of interest if we divide the square of the hypotenuse by ten we obtain five which is also the length of the side of the square. And if we obtain the square root of five we get two point two three six zero six seven nine seven seven four nine nine seven nine.
Adding one to the square root of five gives 3.236 067 977 499 79.
And dividing that by two we obtain the golden ratio which is: One point six one eight zero three three nine eight eight seven four nine. Or 1.618 033 988 749.
Since the radius of the circle is also one of the other two sides if the side of the square is the hypotenuse of one of the four smaller triangles. And the radius of the circle is three point five three five five three three nine zero five nine three two seven three eight. Or 3.535 533 905 932 738. Then if we find the square of that we obtain twelve point five. Which means that 12.5 is the square of each of the other two sides in our right angle triange. 12.5 + 12.5 = 25. So the hypotenuse of the triangle must be the square root of twenty five which is five.}}}
Rather than correcting the present argumentation up to this stage with a view to clarifying some statement of my terms since it is possible that my correction would be less satisfactory than the present statement. Then I would say that as stated the argumentation up to this point seems to make sense in the way I intend it to make sense. And the information seems correspondent to what I want it to refer towards. Therefore, what I would rather do is provide a less explanatory model that defines the key matters we should be clear in regard towards, without confusion, grey areas, blanks or error. It does seem to me that there can be no progression of mind or intellect beyond these early matters until these early matters are clearly stated in necessary sufficient terms without error and without confusion. By which I mean there seems little point in attempting what can only be more difficult and more sophisticated until we are quite confident about our understanding of those prime areas on what the more difficult sophisticated matters are based.
Although of course that is not quite the reality. Rather it is that the mind intellect capacity to coherently inform as to the prime patterns involves a confidence that may only be obtained by beginning with what seems more difficult and sophisticated. It is somewhat as Aristotle says when he supposes we begin any study of the principles and elements of some subject matter by looking at what is more general and evident to our senses and less true, and use that as our entrance towards what is more abstract, universal and less evident to our senses, and more true. Therefore in that way the idea would be to perhaps neither lift up nor intend to move deeper when we find a relatively stable basis at which we can educate ourselves as to principles and elements of geometric relationship in an ideal form.
Because if we travel deeper we would be moving out of the present subject as fast as we entered it and find ourselves in the class defining the concepts of physics which whilst interesting is not the same as the study of geometric relationship of simple shapes. Or if we lift up we would be moving towards the class applying mechanical formula to levers and pulleys which must be considered as too practical for our purposes. Instead if we maintain our present level we will be under the control of ancient masters such as Pythagoras, Archimedes and Euclid, those three being the founders of this school of thought. And informed by modern day masters such as Kepler, Galileo and Einstein, those three being central to all of our present knowledge base.
Further, I would make the point that at this sort of depth in terms of geometry the entire school is divided into those who make friends with the Elements of Euclid, On the Heavens of Aristotle and Timaeus of Plato and those who do not. Since all three books are very small and could all three together be read in one long afternoon, there is no reason to let them remain unfamiliar, even if nothing else written by the same authors was studied.
To clarify, the model of this system is as follows:
A.
1. Make a point.
2. Extend a line from the point.
3. Bisect the line through its centre.
4. Draw a third line equidistant from the point of the bisection such that the third line crosses each of the extensions of the first two lines and both ends of the third line are connected together.
5. Draw four connecting lines from the point where the first two lines touch the circumference of the circle.
That figure is the circle, square and triangle in relationship. It can be alternately described as "that geometric figure made by the circle circumscribing a square divided by four similar triangles".
B.
1. Go back to definitions. What are better definitions of point, line, plane, object, tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron, sphere.
2. Clarify the nature of each of the defined terms. What can be learned by induction, analogy and anomaly. How are the concepts of similarity, generality, familiarity, constancy each one useful in informing us as to the particulars and universal nature of the terms.
3. Attend towards the conceptual relationship between the elements of the given figure. That is with the intent to exhaust what can be known as to the constant relationship between a triangle of two fourty five degree angles and one ninety degree angle, the four similar such triangles that together make up any square, and the circle whose diameter is the diagonal between two extreme corners of the same square.
C.
1. Consider possible extensions of the same study in any direction and determine what are the subject matter that movement in that direction attend towards, for example, deeper being physics and more practical being mechanics.
2. Consider the science that informs the structure of geometry that being the mathematics of trigonometry, calculus and logic.
3. Notice more tenuous connections such as the musical notation system where the three, seven, twelve relationship is built in. Or the music of the spheres defined by Kepler where the relationship of the five prime solids is used to structure an astromomy. Or the elemental system that is based on the cosmology of Socrates.
Circumference = 22.2 144 146 907 9181
Radius = 3.53 553 390 593 2738
Diameter = 7.07 106 781 186 5475
Quink = 4.44 288 293 815 8362
Side of Square = 5
Square of side of square = 25
Diagonal of Square = 7.07 106 781 186 5475
Sum of square of other two sides = 50
Square Root of Fifty = 7.07 106 781 186 5475
Pi = 3.14 159 265 358 979
1. Given the length of the circumference of any circle.
2. Divide the length of the circumference by Quink to obtain the length of the side of square that the circle circumscribes.
4. Multiply the length of the side of square by four to obtain the length of the perimeter of square.
5. Multiply the length of one side of the square by itself to obtain Square of side of square.
6. Multiply the Square of side of square by two in order to obtain Sum of square of other two sides.
7. Discover the square root of the sum of square of other two sides to obtain the diameter of circle with the original circumference.
8. Divide the original circumference given in note one, by the diameter given in note seven, to obtain Pi.
For a complete index to the various articles I have used to introduce these and related patterns, please follow the hyperlink:
