Chromogeometry is a quite beautiful development from rational trigonometry.
At 42 minutes 57 seconds Wildberger talks about the groups associated with these three covariant geometries in the same vector space. See:
See this video on fields, which are in some sense pairs of groups, with a linking relation (distributivity) between their operators. In particular, see the part at 5 minutes 53 seconds on finite fields, prime fields and Characteristics.
Now see this lecture on abstract vector spaces, which are in some sense pairs of groups, one of which is a commutative group of vectors, and the other of which is a field, which as we saw above, is itself a pair of groups. A vector space also has a linking relation between the group operators, which is associativity.
If we relax the condition of multiplicative inverses then the field becomes a ring, and we have a more general abstract space called a module. See in particular the example at 3 minutes 54 seconds, where the module is the set of real 3-vectors, and the scalar ring is the set of linear transformations of that space.
Now watch this video, which shows how an algebraic "language" called an Iterated Function System, or IFS can give a geometrical interpretation of the real numbers in the open unit interval.
There are some interesting connections here with Asymmetric Numeral System or ANS Encoding. See Asymmetric numeral systems, which gets us to probabilistic models of numerical computation which have potential as a foundation for numerical methods for solving differential equations.
At 45 minutes 41 seconds there is a question about chromogeometry in three dimensions. An interesting place for ideas on this would be Charles Dodgson's book An Elementary Treatise on Determinants: With Their Application to Simultaneous Linear Equations and Algebraic Geometry, chapter VII in particular:
See from 9 minutes 4 seconds and from 14 minutes 40 seconds for an explanation of polarity, and a way of calculating tangent points to conics.
Here is another geometrical characterization of the open unit interval, called the Stern-Brocot tree, which also associates a rational to every string of symbols from an alphabet of two elements.
Here is another way to define the Stern-Brocot tree as a sequence with the successive terms defined by an Iterated Function Sequence.
The Stern-Brocot tree has a very neat characterization as an iterated transformation of contraction mappings characterized by 2x2 matrices of positive integers. These matrices each have a determinant of 1 and correspond to triangles of area one half.
This construction seems to be enough to get chromogeometry off the ground.
See from 5 minutes 37 seconds of this video on units in a ring, which gives the example of 2x2 matrices with integer entries
Then see this video on harmonic conjugate ratios:
... and some interesting properties which are connected with the Golden ratio phi and the Stern-Brocot tee:
I'll try and explain better what I mean tomorrow, but it is a connection between multiplication and addition which allows one to express solutions to an equation like n=n+1 in a useful way. It is the "nagging issue" Wildberger talks about at 33 minutes 41 seconds and which Jean-Yves Girard calls The Skeleton in the Closet. This theorem of the cross-ratios is, I think, the essence of proposition 16 of Euclid book V: Proportionals Alternate.
If you are wondering what an Euler line is, it is a definition of a remarkable property of any triangle explained by Wildberger in this short video. It is that in any triangle, the orthocenter, the centroid and the circumcentre are colinear and they divide the line in the ratio 2:1.
At 28 minutes 45 there is a cute characterization of an ellipse in terms of four foci and four directrices. Recall this interesting way to look at the stability of sequences representing numerical solutions to things like the logistic equation (see The Weirdest Math Video You'll See Today):
See this excellent video
Then watch this, particularly the bit about eigenbases at 13 minutes 3 seconds.
At 42 minutes 57 seconds Wildberger talks about the groups associated with these three covariant geometries in the same vector space. See:
See this video on fields, which are in some sense pairs of groups, with a linking relation (distributivity) between their operators. In particular, see the part at 5 minutes 53 seconds on finite fields, prime fields and Characteristics.
Now see this lecture on abstract vector spaces, which are in some sense pairs of groups, one of which is a commutative group of vectors, and the other of which is a field, which as we saw above, is itself a pair of groups. A vector space also has a linking relation between the group operators, which is associativity.
If we relax the condition of multiplicative inverses then the field becomes a ring, and we have a more general abstract space called a module. See in particular the example at 3 minutes 54 seconds, where the module is the set of real 3-vectors, and the scalar ring is the set of linear transformations of that space.
Now watch this video, which shows how an algebraic "language" called an Iterated Function System, or IFS can give a geometrical interpretation of the real numbers in the open unit interval.
There are some interesting connections here with Asymmetric Numeral System or ANS Encoding. See Asymmetric numeral systems, which gets us to probabilistic models of numerical computation which have potential as a foundation for numerical methods for solving differential equations.
At 45 minutes 41 seconds there is a question about chromogeometry in three dimensions. An interesting place for ideas on this would be Charles Dodgson's book An Elementary Treatise on Determinants: With Their Application to Simultaneous Linear Equations and Algebraic Geometry, chapter VII in particular:
See from 9 minutes 4 seconds and from 14 minutes 40 seconds for an explanation of polarity, and a way of calculating tangent points to conics.
Here is another geometrical characterization of the open unit interval, called the Stern-Brocot tree, which also associates a rational to every string of symbols from an alphabet of two elements.
Here is another way to define the Stern-Brocot tree as a sequence with the successive terms defined by an Iterated Function Sequence.
The Stern-Brocot tree has a very neat characterization as an iterated transformation of contraction mappings characterized by 2x2 matrices of positive integers. These matrices each have a determinant of 1 and correspond to triangles of area one half.
This construction seems to be enough to get chromogeometry off the ground.
See from 5 minutes 37 seconds of this video on units in a ring, which gives the example of 2x2 matrices with integer entries
Then see this video on harmonic conjugate ratios:
... and some interesting properties which are connected with the Golden ratio phi and the Stern-Brocot tee:
I'll try and explain better what I mean tomorrow, but it is a connection between multiplication and addition which allows one to express solutions to an equation like n=n+1 in a useful way. It is the "nagging issue" Wildberger talks about at 33 minutes 41 seconds and which Jean-Yves Girard calls The Skeleton in the Closet. This theorem of the cross-ratios is, I think, the essence of proposition 16 of Euclid book V: Proportionals Alternate.
I've wondered about how chromogeometry could become 3 dimensional. The idea I have come up with is seeing the 'familiar' Blue, Red and Green unit circles as sections of right-cricular cones clustered in 3-space. The Blue circle is then the cone: x^2 + y^2 - z^2 = 0 viewed 'in' the plane z = 1. The Red cone is either postive quadrance (section: x^2 - y^2 = 1, cone: x^2 - y^2 - z^2 = 0) or negative quadrance (section: x^2 - y^2 = -1, cone: x^2 - y^2 + z^2 = 0). The Green cone(s): 2xy -/+ z^2 = 0.
ReplyDeleteIn addition, symmetry requires the inclusion four additional cones at 45 degrees to the xy plane for a total of 9 cones.
When viewed from different facets of the cube the orginal 'circles' assume alternate identities. When viewed from 'intermediate' (isometric) projections the 'circles' are simply different conics. The presence of additional 'circles' in any planar view might be a way of relating triangles in different projected planes.
(Maybe '3d'? But it would still be projective - albeit in 3 axes.)
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