the role of number bases in future math?
kirby urner
is the notion of number base indeed belongs
in any common core curriculum, so that at
least bases 2 and 16 may be highlighted,
followed by a re-visiting of calendrical time
as a mixed-base system.
This recent movie 'The Martian' starring
Matt Damon, makes a point of emphasizing
that the hexadecimal encoding of the
alphabet and other keyboard characters,
known as ASCII, is core heritage within
NASA and JPL at least, something any
would-be astronaut should know about.
ASCII is a doorway into the whole world
of "bits & bytes" i.e. base 2, starting from
a "C.P. Snow chasm-spanning bridge" in
miniature -- said chasm being Snow's
metaphor for the gulf twixt humanities and
science languages, a divide leading to
of just "base 10 arithmetic" and it is, most
assuredly. We might use the inclusion of
Unicode as a hallmark of a future math.
In talking about how "bits & bytes" map
to large numbers of symbols, such as
Chinese characters, we're getting to
important mathematical concepts such
as "mappings" and "sorts" (collations).
The alphabet gives us opportunities to
talk about the operators > and <. The
relevant to future math.
UTF-8, the most used Unicode encoding,
is a mapping of symbols to sequences of
1s and 0s.
"mixed base" is ripe for exploration. All
these topics belong in a spiral pattern
that includes both logic (computation)
and lore (history).
How seconds and minutes link to distance
as well, through Earth's rotation, lines of
longitude, returns us handily to the geometric
domain and its spheres and circles, including
unit radius spheres and circles. The link twixt
duration and spatial extent, between the
temporal to the spatial, is intrinsic to STEM.
The concept of "light year" takes us into the
namespace of (the realm of) astronomy.
of number bases as a topic, to any purportedly
STEM-friendly common core.
Bradford Hansen-Smith
Kirby, you said, “the geometric domain and its spheres and circles.” My experience suggest maybe a sphere/circle domain and its geometry. Base 10 is part of a 0-9 sequence. We usually do not start with the zero, defined as nothing; nothing to go out from an empty set. (The old measurement question; is one the interval between nothing to number or is number one where we start, or it there another possibility?) Understanding the circle zero is also origin that reveals 0-9, then first frequency forward starts with 10. By compounding frequencies in sets of 1-9 through systematic combining and rearranging sequence of numbers we discover amazing relationships with great breadth about time and space. The circle is origin because it inherently carries the dynamics of spherical compression. This is not about the concept of a slice through a sphere that we draw a picture, call empty, thereby disregarding unity of the spherical whole. The static circle images a sphere to circle compression, a dynamic object that when folded decompresses spherical information.
The first fold of the circle in half is a 1:2 ratio showing 0-1-2-3 number sequence. That one fold generates two tetrahedra, one rotationally reciprocal to the other. Touching any two points on the circumference generates two more points, making four points in space with six relationships between them defining four planes, two are open and two solid. The primary number of points and lines that define four planes is 10. This is one frequency up from unity to the folded circle where the diameter and circumference have not been separated from origin. The dynamic transformational nature of the circle is context from which numbers are abstracted and relationships isolated. If this information was not first in the circle/sphere, they could not be identified and constructed.
Given the tetrahedron as first fold is common to all symmetries and pattern based formations, this information belongs in all common core and other math curriculum. ASCII coding is not fundamental but a higher level of abstract language that makes it easier for mechanical manipulation. The hexadecimal encoding is important because it embodies the first three proportional 1:2 folds in the circle. Three equally spaced diameters generate seven points (hexagon pattern) that when taken together in the circle is the number 10, divided by 2 gives us 5, a compacted number representing the primary properties of a single tetrahedron.
It would make sense that UTF-8 is most used for coding 1’s and 0’s. Eight is the primary octave division of the 3 circle diameters of 9 points each. The number 27 ends the third frequency set from which follows sets of 1-9 each in turn revealing 8 intervals. Not much different than music, nothing happens without the dynamics of space and time between the nodes. The concatenation (A + B = AB) is relevant to future math only to the degree that we find a relative healthy balance between virtual reality experiences and direct human experience. The first fold is complimentary movement in two directions and suggest two exist as duality in relationship which by definition is three; 1 in isolation does not exist. I only bring this stuff up because we need to look at all the options.
Brad
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kirby urner
Your reminder that the first folding of the circle (squashed sphere) brings back our memory of a full sphere, with that sweeping action.
Your first fold reminds me of Koski's "one (or two) page triangular book":
The book covers lie flat (book is wide open) but they're equilateral triangles, not rectangles. One triangular page flaps back and forth through an arc of 180 degrees, from lying flush against one cover to flush against the other.
We may use this device to measure the volume of the resulting complementary tetrahedrons, connecting page tip to each cover tip.
Thing is: Earthlings (E) and Martians (M) set their respective "unit volume" at a different place:
Pictures:
https://flic.kr/p/fwsARu (one page triangular book)
https://flic.kr/p/fwdt8t (page straight up volume = volume of R-edge cube, R = 1/2 page edge length).
Your website explains it, and then runs ahead with Tetrahedron and Octahedron, in ways I'm also somewhat well versed in (tetrahelix etc.).
Excellent! And all with just one paper supply (and some tape)!
>>> for chr in 'ABCDEFGHIJKLMNOPQRSTUVWXYZ':
... print( "{} = {} = {}".format(chr, "{:#08b}".format(ord(chr)), ord(chr)))
...
A = 0b1000001 = 65
B = 0b1000010 = 66
C = 0b1000011 = 67
D = 0b1000100 = 68
E = 0b1000101 = 69
F = 0b1000110 = 70
G = 0b1000111 = 71
H = 0b1001000 = 72
I = 0b1001001 = 73
J = 0b1001010 = 74
K = 0b1001011 = 75
L = 0b1001100 = 76
M = 0b1001101 = 77
N = 0b1001110 = 78
O = 0b1001111 = 79
P = 0b1010000 = 80
Q = 0b1010001 = 81
R = 0b1010010 = 82
S = 0b1010011 = 83
T = 0b1010100 = 84
U = 0b1010101 = 85
V = 0b1010110 = 86
W = 0b1010111 = 87
X = 0b1011000 = 88
Y = 0b1011001 = 89
Z = 0b1011010 = 90
(A) we expand the notation we share with children to include the "dot" (as in "dot com" and "dot edu"). Noun.Verb( ) and Noun.Adjective will become more common as integral with arithmetic operators.
In other words, addition is an operation (with number types), but so is concatenation (with the character type), and both may use + (the symbol). Godel Escher Bach helped with bringing in the string type (= characters) whereas we all need to understand the overloading of symbols (the importance of context is key, meaning stems from usage patterns).
In the context of (C) we'll be focusing more on (B) (A) and (D). True, some of this stuff looks a little unfamiliar, especially the "dot stuff", but we're still happy to call it M as in MEST (STEM) or whatever.
Many texts will inscribe or circumscribe, such that the tetrahedron and single sphere are concentric, the the former's tips just touching the sphere's skin from the inside, or else the sphere is tangent to the tetrahedron's four facets from its inside.
I'd like to make sure we keep: four unit-spheres in intertangency, with a tetrahedron defined by edges connecting the four sphere mid-centers. That's an image we need to spend time with as well, especially in our Martian Math curriculum.
Bradford Hansen-Smith
Kirby, you ask, “What's a relationship between Tetrahedrons and Spheres?” followed with a text book generalization. As you know tetrahedrons can best be understood by hands-on experience, as you mentioned, “four unit-spheres.” While our experience is putting things together, a summation process is not what is going on. A single sphere when it divides and multiples forms a tetrahedron pattern of multiples of itself showing six points of divisional connection that retains unity of the single sphere. There has been no separation yet an inner/outer space has been formed. Similar to how one might conceive of forming a torus from a sphere without breaking the surface of the sphere. An extraordinary thing, as common place as conceiving children. If we do not start looking towards “Future Maths” from a more comprehensive place we will never get out of an infinite number of fragmented ways of thinking about it.
Leave the triangles and tetrahedra in the circle, there will be more information to work with, more things to look at, more questions to ask, and more multiple relationships become obvious without having to create separation.
The primary unit volume is the sphere, the triangle and tetrahedron are secondary. Here are some questions for your students. What is the volume of the compressed sphere given the diameter and thickness of the circle in their hands? Have them guess, then calculate the difference; this is always a surprise. What is that initial spherical volume in relationship to the increased volume of the circle folded in a spherical pattern? What is the volume of the tetrahedron formed in that fold? Does it change with degrees of openness? Koski's demonstration. How does the volume of the tetrahedron (or two) compare to the volume of the folded sphere, to the original compressed sphere? By creating interest in volumetric relationships then the means to find out becomes secondary to purpose. Math should be the means not the purpose.
As the one video clearly shows we are “overloading of symbols” and you are right, only through understanding pattern as the first structural happening can we establish a meaningful context in which to systematically make sense of endless pixels, parts and pieces. Until that happens there will be no Furture Maths, only the same over and over in different forms without understanding the nature of spherical unity as the only inclusive context that brings together what has never been apart, except in our perception from an ego-centered, geo-centered position.
I had a go at a visual triangle book in the past. It might interest you and can be seen at the end of this clip.
https://www.youtube.com/watch?v=-pUU5h-oCbE
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kirby urner
Kirby, you ask, “What's a relationship between Tetrahedrons and Spheres?” followed with a text book generalization. As you know tetrahedrons can best be understood by hands-on experience, as you mentioned, “four unit-spheres.”
https://youtu.be/qnk4f6bsIA4
http://etc.usf.edu/clipart/42200/42256/sphereinscrb_42256_lg.gif
(C) a sphere tangent to tetrahedron mid-edges:
http://liris.cnrs.fr/~rchaine/IMAGES/sphere_tetrahedron.jpg
https://quadralectics.files.wordpress.com/2013/09/306.jpg
While our experience is putting things together, a summation process is not what is going on. A single sphere when it divides and multiples forms a tetrahedron pattern of multiples of itself showing six points of divisional connection that retains unity of the single sphere. There has been no separation yet an inner/outer space has been formed. Similar to how one might conceive of forming a torus from a sphere without breaking the surface of the sphere. An extraordinary thing, as common place as conceiving children. If we do not start looking towards “Future Maths” from a more comprehensive place we will never get out of an infinite number of fragmented ways of thinking about it.
https://youtu.be/tgKC-awk6p4 (CCP vs. HCP)
http://www.grunch.net/synergetics/images/rdpack.gif (3-frequency cuboctahedron -- 92 spheres in outer layer)
http://synchronofile.com/wp-content/uploads/synergeticsfigure2201c1997.gif
https://oeis.org/A005901
http://www.4dsolutions.net/ocn/graphics/sphjit.gif
Leave the triangles and tetrahedra in the circle, there will be more information to work with, more things to look at, more questions to ask, and more multiple relationships become obvious without having to create separation.
The primary unit volume is the sphere, the triangle and tetrahedron are secondary. Here are some questions for your students. What is the volume of the compressed sphere given the diameter and thickness of the circle in their hands? Have them guess, then calculate the difference; this is always a surprise. What is that initial spherical volume in relationship to the increased volume of the circle folded in a spherical pattern? What is the volume of the tetrahedron formed in that fold? Does it change with degrees of openness? Koski's demonstration. How does the volume of the tetrahedron (or two) compare to the volume of the folded sphere, to the original compressed sphere? By creating interest in volumetric relationships then the means to find out becomes secondary to purpose. Math should be the means not the purpose.
Starting with 2 rods at 60 degrees with a common vertex O, and modeling A x B as the area AOB is a 2nd powering illustration i.e. AOA = area A * A.
Bradford Hansen-Smith
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kirby urner
Thank you Kirby for the video. Yes it is the same, with a couple of procedural differences. Zander is a kid I met about about twelves years back when his mother bought a book about folding circles. The instructions at the end of the video were from a handout paper I did at that time. It is nice to see young people make folding circles their own and then to teach others. Isn't that the point of all learning? Teaching is then the passion you do that stimulates and moves students to do something on their own. Why must we train young people to be students of learning when learning comes naturally at birth. It must be obvious, I have no training as a teacher, or in math for that matter.
Thanks for clarifying Martian Math. Like Zander, we all take on what we understand about what we learn, is always different than the source and that is what often leads to confusion that generates openings for clarification and new ways of thinking about things. Martian Math sounds alien and what you are talking about is clarifying a system of inconsistencies made over centuries of logistical errors. Yes I see your point, from a position of being here any there is always going to be alien.
"2 cubed" sounds to Martian ears (yes they have ears) more like a school-conditioned reflex, never questioned, with the "most schooled" tending to be the "least open" to Martian Math.
Maybe that's why the hydro-power dam idea proved more productive, as an outdoor "get work done" activity. Over time, we can hammer out some better bridges **, in terms of curriculum, for those workers needing to comprehend both cultures.
kirby urner
Yes, Martian Math is quite alien, in departing from the cube as the only logical model of 3rd powering. It's so ingrained in Earthling culture to refer to 2 to the second power as "2 cubed" and yet for Martians, a tetrahedron with 2 on each side, and volume 8, is what they would call logical.
Maybe that's why the hydro-power dam idea proved more productive, as an outdoor "get work done" activity. Over time, we can hammer out some better bridges **, in terms of curriculum, for those workers needing to comprehend both cultures.
We show about the importance of Ortho to the Martians by holding up a tetrahedron and tracing out either of two 3-edge zig-zags that comprise it. Z + N = tetrahedron (the edge 3 coming off the paper at a slant).
http://www.rwgrayprojects.com/synergetics/s01/figs/f0801.html
In Martian Math, they put a lot of stress on the fourness of the simplex, the minimum polyhedral convexity / concavity. Whereas Earthlings are sure Kantian space (same as Euclidean) is "three dimensional", we translate Martians as saying it's "four dimensional" (four vertexes, four windows, two 3-edge zig-zags) with no way to break it down into fewer atomic dimensions.
All their primitive entities are "lumps" i.e. their "points, lines and planes" are distinguished by shape, not dimension.
Fortunately an Earthian named Karl Menger, a Vienna Circle mathematician, later moved to Illinois (I know his daughter Eve) came up with that same idea -- a "geometry of lumps" -- so we have a bridge there. The essay wherein he spells that out is: 'Modern Geometry and the Theory of Relativity', in Albert Einstein: Philosopher-Scientist , The Library of Living Philosophers VII, edited by P. A. Schilpp, Evanston, Illinois, pp. 459-474.
The Martians haven't sent any "kids" to our planet yet as to them, the Earth is regarded as "for adults only" at the moment ("no minors allowed"). They have like a rating system for planets, roughly similar to our Hollywood rating system: G, PG, TV-14, NC-17, R, MA. They think of Earth as R-rated and above, one might say.
But for Earthian kids, we can easily keep the curriculum all safely G and PG (though for older kids we may choose not to). The Earthians are getting a lot out of this collaboration and are spreading Martian Math memes around the globe, despite all the reflex conditioning and schoolish brainwashing.
Christian Baune
When I was young we learned to count up to 1024 on fingers.
We learned how to multiply using fingers.
We also had the Cuisener rodes to understand how a positional system work.
Sadly, this has been removed in European curriculum, but it was there.
A lot of problems have a simple solution: roll back to old cursus.
Learning bases is essential to understand how we worked out our numerical system.
kirby urner
When I was young we learned to count up to 1024 on fingers.
We learned how to multiply using fingers.We also had the Cuisener rodes to understand how a positional system work.
Sadly, this has been removed in European curriculum, but it was there.
A lot of problems have a simple solution: roll back to old cursus.
Learning bases is essential to understand how we worked out our numerical system.
He was the founder of the British equivalent of the NCTM, the ATM:
http://www.atm.org.uk/Caleb-Gattegno
http://www.bbc.com/news/technology-31834927
Teaching only base 10 is uber-parochial (in the sense of narrow minded). One doesn't really understand what "base 10" even means without exposure to alternatives. How does one "compare and contrast" with only one example? Teaching only "base 10" is teaching no bases at all, it's eschewing the whole concept. I think it's safe to call any "base 10 only" curriculum a "dead end" as in "not long for this world".
http://www.nytimes.com/2015/09/16/nyregion/de-blasio-to-announce-10-year-deadline-to-offer-computer-science-to-all-students.html?_r=1
Bradford Hansen-Smith
What you are saying about Martian Math is true but it goes deeper than units of measure, functional practicality and consistency of pattern, it goes into philosophical differences that plague the world we live in. Math is not liked by many people, hated by some, because it is "difficult," "hard to learn." It goes deeper than that, it is ubiquitous. Math has permeated out culture and much of the world with a logic that is flawed, fragmented, and without principle or structural sustainability. When will people that love math take some responsibility and question what we are doing and when will the rest of us figure out it is not the math that is difficult.
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kirby urner
kirby urner
Kirby, first, we are all autodidacts until we are put into school and directed away from inside to the outside ourselves which means having to learn a body of predigested-by-others knowledge, not to be questioned, only tested for easy and profitable recovery, nutrition is not required.
Just to advocate for the devil a bit (we get along :-D):
We do need attention directed away from the inside, lest Johnny and/or Sally become a complete introverts. People who nerd out in the extreme and don't have social lives are rarely successful as ball players or cheerleaders. Who will serve us food in the cafeteria if we don't pass on the predigested-by-others culture, an investment of some thousands of years?
You may question, but don't waste everyone's time. Teachers are busy people.
That is what we are up against. Observationally the tetrahedron is primary to the cubic system, even in solid form we can see that.
Why isn't V + F == E + 2 part of the CC (Common Core) standard in the US? I'm glad we have other smarter CCs that are not made in the USA. Euler's stuff features in a Bhutanese mathematics curriculum at least -- I know cuz I wrote it.
The tetrahedron with 2 on each side is an acknowledgment of spherical origin.
Anyway, check the classroom poster. You learned all this in 3rd grade (OK, maybe not if you were born pre 2000, but those were dark times :-D).
This suggest a one-frequency tetrahedron is a functional generalization missing a lot of information for the sake of this idea about making it simple or easy, which is a concession to teaching, while at the same time devalues natural self-learning and the capacity of students.
Not that any Earthling agency or entity has a monopoly on MM, a thin coating over the Bucky stuff, a distillation (like a Scotch). I'm free to share it in the RSA (Republic of South Africa), which tends to be more open-minded than the USA about a lot of stuff. Algebra City...
There is only one closest packed order of spheres and all the others are favored sets that are plucked out as individualized and truncated systems. It doesn't take long to forget they all come from the same ordered multiplicity of spherical unity. Once in separation we argue over which is more important than the other.
They're patterns without substance at first, like XYZ. We don't care of the loci are comprised of atoms or molecules. We don't care if there's any material in the picture at all. They're "skeletons" or "scaffolds" or "frameworks".
When we take all the CCP balls and inter-connect adjacent sphere centers, we get a framework known as the IVM in many walks of life (short for "everywhere-the-same vector-edges" or "isotropic vector matrix"). XYZ and IVM go together, as alternative scaffoldings. Earthlings have tended to prefer XYZ whereas Martians are IVM-oriented. De nada, they both make sense and have utility.
Compression along any axis of sphere to circle disk is a right angle function. The first fold in the circle is a right angle function where the resultant crease is perpendicular to the distance between any two points, a spherical movement. The relationship between centers of the "empty" space in the packing of spheres is cubic; the seven axis of the tetrahedron are right angles to and from the planes and edge lines.
http://www.rwgrayprojects.com/synergetics/s09/figs/f87230.html
Right angle precessional movement is seen through eight sectors of the xyz axis and is primary relationship between into/out duality.
XYZ axis may be abetted with Q-rays i.e. four vectors emanating from O dividing space into four quadrants instead of eight XYZ octants. All positives numbers along each ray (reserving negatives for a "dual space").
CCP centers have all integer coordinates when we calibrate it that way i.e. I keep the Q-ray tetrahedron edges as 2R (=D), meaning the "unit vectors" from (0,0,0,0) to (1,0,0,0) (0,1,0,0) etc. aren't really unit length.
Lets call 'em "basis vectors" anyway, all four of 'em. Every point in space is reached by a simple sum of these four. Call this apparatus 4D if you like (the Martians do). Then add energy for more dimensions.
Only when represented in 2-D is this angulation static and fixed at 90 degrees which has become standard for calculations of all kinds. How much more interest and information there is when we drew out concentric circles giving back context to the xy axis. My point being we do not know how to get out from under the magnitude of errors and distortions that has accumulated over centuries developing a logic system based on 2-D distortions of 3-D forming in a 4-D reality.
By posing a hypothetical ET civilization that is already free of our biases, never having suckered into them, we have a better chance of studying IVM versus XYZ as *ethnography*.
Anthropology (the A in STEAM) allows for cool-headed, methodical, strategic approaches to inter-lifeform dialog.
Our secret "collaboration mesa" (where we're building the hydro-dam with the Martians, per storyboards) is protected against incursion by the angry dogmatists who can't stand the idea of Earthian children questioning their authority, now that the Martians have landed.
There's a LuxBlox [tm] fence to protect the Martian apartments: https://flic.kr/p/zCJpe6
We introduce movement into flat images to virtually represent reality. First by moving the pictures in sequence, now in line with that idea of getting static images to move we use cutting edge technology; the abstraction of 1 and 0, the alpha and omega of base ten, complete denial of 0-9. Can the fault line be more obvious?
We're into stereography more and more e.g. with those colored glasses. Holographics etc.
Descartes proves relevant in talking about Res Extensa. It needn't be energetic i.e. we can imagine a purely imaginary "holodeck" where "nothing exists" (like a perfect vacuum). That's pre-energetic and we only go there in our dreams.
[ Martians talk about dreamtime a lot and enjoy didgeridoo concerts (interesting factoid!) ]
Your comment about the movement of the tetrahedron and the two three-edge helices brings up the video of Zander folding the circle. If you watch, after the first fold he eyeballs the second and makes the third fold on top of the second, hiding from view what ever accuracy or inaccuracy there is. Had he folded the third section to the opposite side all edges and points would be visible making it easy to gauge any inaccuracies. In doing that the folds forms a "z", one over and one under the middle third. The three double edges are six edges with six triangle planes forming two tetrahedra each with two solid planes and two open planes. This is not coincidence that the first fold in half shows the same pattern of two tetrahedra, each with two solid planes and two open planes, albeit there is a change in scale and regularity of triangles.
So is using a protractor OK? He brings that up in the beginning and suggests some of his teachers were discouraging of using one.
Brad
What you are saying about Martian Math is true but it goes deeper than units of measure, functional practicality and consistency of pattern, it goes into philosophical differences that plague the world we live in. Math is not liked by many people, hated by some, because it is "difficult," "hard to learn." It goes deeper than that, it is ubiquitous. Math has permeated out culture and much of the world with a logic that is flawed, fragmented, and without principle or structural sustainability. When will people that love math take some responsibility and question what we are doing and when will the rest of us figure out it is not the math that is difficult.
In tandem with hexagons.
[ Picture of soccer ball goes here (Martians love soccer, even if their ball is "imaginary" (they have sports in shared dreams -- not something Earthlings know how to do without television) ]
Bradford Hansen-Smith
Ah Kirby, to play the devils advocate is to compliment the devil for getting you to do his work. It is said the devil is in the details, what ever the hell that means. My guess is de-tail is what drives the cart, lug nuts aside.
Yes, at the very least our topographical reading of polyhedra and of people are mostly focused on outside concerns without much regard for the inner. I might go so far as to say the inside is the feminine and the outside masculine given your assignments as they relate to the world-wide political climate.
Euler's formula P + A = L + 1 applies to any line drawn image and for that reason alone math should include the visual arts. The added constant (quantitative difference between points and a line) when combining separated parts of a drawing or unattached sets in math, will increase as the number of related parts or sets increase. Start simple by drawing a smiley face (a circle head, two dots for eyes and a line for the mouth) and apply the formula, One point where the circle starts and ends, plus two eyes, one circle area, adding four independent parts that make up the image; 6 = 6. How beautiful that a math formula can be used to collate all points, lines, and areas for any line drawn image, from Rembrandt to Matisse. Knowing that why would we want to? As for V + F = E + 2, it works for the same way for any collection of independent objects in space; the stereographic path to virtual wonders. But it is not consistent to all objects, therefore not as comprehensive as we think. It does not apply to a circle disc.
While also understanding the six symmetrical divisions
of the tetrahedron/octahedron is important, I was referring to the 7 axial
rotations of the tetrahedron where 3 pairs of opposite edges are at right
angles and each vertex is at right angles to the center of the opposite face. The
point being that the right angle is primarily movement and happens first with
spherical compression.
Staying with the tetrahedron for a moment, It is a pattern that does not have to have the properties we associate with the form. A “solid” tetrahedron can be formed with one crease in the circle that shows four points, three edges, and one surface. The properties are not the same as the straight edge, flat plane form we are familiar with, but it does satisfy Euler’s formula for polyhedra. We have come to believe straight lines and curved lines are not congruent, which is true as far as it goes, but not true when we go further. :-D (6+1 = 3+4) Maybe instead of typing out :-D just type 7 , it's more math friendly. I leave you with that my deviled friend.
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kirby urner
Ah Kirby, to play the devils advocate is to compliment the devil for getting you to do his work. It is said the devil is in the details, what ever the hell that means. My guess is de-tail is what drives the cart, lug nuts aside.
As for "devil's advocate" I understand that was a paid role: the priest sent out to see what contravening evidence might subvert / derail so-and-so's eventual canonization, i.e. the devil's advocate was the hired skeptic, standing between a soul and sainthood.
Arguing as a sport. I'm not against it. To me, that's what diplomacy boils down to, an alternative to violence.
Yes, at the very least our topographical reading of polyhedra and of people are mostly focused on outside concerns without much regard for the inner. I might go so far as to say the inside is the feminine and the outside masculine given your assignments as they relate to the world-wide political climate.
But we want every lug nut to fit every car (I don't know that they do, probably not). Schooling is likewise about passing on a culture such that the humans within it stay inter-operable. If we each get left alone to "construct our own reality" then will industry happen? Will we have iPhones?
We need standards, which is to say we need people to agree on specific protocols. In computing, a standard is not so much about keeping cockroaches out of the kitchen (health and sanitation) as having any means to communicate in the first place. Without standards, there's no way to get a message from A to B. Enter ASCII, enter Unicode. Hexadecimals. Lore.
Euler's formula P + A = L + 1 applies to any line drawn image and for that reason alone math should include the visual arts.
The added constant (quantitative difference between points and a line) when combining separated parts of a drawing or unattached sets in math, will increase as the number of related parts or sets increase.
Karl Menger's idea was we could make such a "geometry of lumps" axiomatic. We simply choose definitions wherein points, lines and planes are all "made of dough" but they can be so thin and/or tiny we have no measure that matters in a specific context.
We thereby dispense with points of "zero dimensions" in the absolutist Euclidean sense. Karl's proposals need to be categorized as "non-Euclidean".
How important was it that lines "really" go to infinity?
Make every line a segment (of a large radius circle maybe -- railroad tracks only look straight as the bend over the horizon), every plane a pancake (or paper plate), every point an "atom" (relative to the challenge at hand).
We don't need these things to be "infinite" -- that was never a requirement for the up close reasoning in Euclid's stuff either, but I'm not proposing any changes to the Euclidean stuff. Leave it as is. We're building a different sandcastle on a different patch of beach.
Start simple by drawing a smiley face (a circle head, two dots for eyes and a line for the mouth) and apply the formula, One point where the circle starts and ends, plus two eyes, one circle area, adding four independent parts that make up the image; 6 = 6. How beautiful that a math formula can be used to collate all points, lines, and areas for any line drawn image, from Rembrandt to Matisse. Knowing that why would we want to? As for V + F = E + 2, it works for the same way for any collection of independent objects in space; the stereographic path to virtual wonders. But it is not consistent to all objects, therefore not as comprehensive as we think. It does not apply to a circle disc.
While also understanding the six symmetrical divisions of the tetrahedron/octahedron is important, I was referring to the 7 axial rotations of the tetrahedron where 3 pairs of opposite edges are at right angles and each vertex is at right angles to the center of the opposite face. The point being that the right angle is primarily movement and happens first with spherical compression.
Yes, I understand about those 7 axes of rotation and was about the mention them myself when I got side-tracked into thinking about A modules and wanting to talk about them.
That sent me on a foray. I don't always know how it will go when I start to formulate my reply.
Staying with the tetrahedron for a moment, It is a pattern that does not have to have the properties we associate with the form.
Indeed. In a way, any four objects: this coffee cup, that dog, my chair, me, form a tetrahedron of relationships, six pairs, of each to the other three.
I relate to the dog, chair and coffee cup. The cup relates to the dog and chair as well. 3 + 2. Then the chair-dog edge brings it to six relationships (3 + 2 + 1).
N things relate N(N-1)/2 ways and in this case, N=2, E=6. When N=2 that's just one relationship.
These would be the triangular numbers, from which tetrahedral numbers may be derived (first two columns of Pascal's Triangle -- another Grand Central Station for us (where many train tracks converge)).
A “solid” tetrahedron can be formed with one crease in the circle that shows four points, three edges, and one surface. The properties are not the same as the straight edge, flat plane form we are familiar with, but it does satisfy Euler’s formula for polyhedra. We have come to believe straight lines and curved lines are not congruent, which is true as far as it goes, but not true when we go further. :-D (6+1 = 3+4) Maybe instead of typing out :-D just type 7 , it's more math friendly. I leave you with that my deviled friend.
Geodesic spheres; radomes... these topics all feature in Martian Math.
Earthlings do seem pretty sheep-like and afraid to question authority. The secret collaboration undertaken with NASA's help, tends to attract the more rebellious who never entirely succumbed to Earthian orthodoxies. Fundamentalists dogmatists have a harder time giving any air time to Aliens.
Bradford Hansen-Smith
Kirby, Thanks for the research on "devil's advocate." And insight about your serious playfulness. I mostly enjoy seeing where your mind goes when a connection is made and the tangents appear as easily as alien spacecraft.
“Codified rebelliousness” is not just a positive direction. There is difference between unity and uniformity. Uniformity as a creativity effort can happen on both sides, progressing towards realized potential or counter towards decay. Either way unity is dynamic interaction between positive and negative where as balance cancels each other out. Conformity when self-imposed for the betterment of all others is forward, otherwise it abrogates individual action.
As for “best schools,” there are models out there than need to be seriously looked at in order to change the pedagogy and curriculum. Change one too much and the other must change. We keep systemic change at a distance by breaking disciplines into smaller areas keeping them separate to manage the illusion of control by making small isolated changes that do not address deeper issues.
iPhones? It goes without
saying, we use a lowercase i and an upper case P, putting emphasis on the technology
rather than the user. Would it be more appropriate if we called it an "eyePhone."
A problem of language standards? Yes we need standards. There is more agreement on technological protocols and building bridges than how people will use them. The cockroaches seem to not have this problem of standards, which is maybe why they go to the kitchen. Where does math look for standards when there are no definable math principals/principles from which to move out from with directives about how to move. Rules and laws codify but do not necessarily suggest standards of behavior.
Clay is a wonderful medium as you have indicated, and I understand the aptness of your battlefield analogy, I’m on your side if there is a battle. But I do not see how you are going to reduce a ball of clay to a smaller size point unless you move far away changing the relative scale or to remove some clay changing the original volume meaning it is no longer the same ball of clay. The material is not the point, it is the transformation of the whole or entire sphere using the clay as a medium. Clay is an additive and subtractive material, where as the sphere is not restricted to any medium and to add or take away denies unity and either way the sphere becomes less than what it is.
Thanks for reference to Edward
Popko’s Divided Spheres. I have not seen the book though know of it.
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