new Martian Math video and blog post
kirby urner
http://controlroom.blogspot.com/2015/10/martian-math-storyboards.html
Bradford Hansen-Smith
Kirby, thanks for sharing your video. I now
better understand how you fit with Fuller. You make good points about why he is
not better known or I should say, not known at all, given my contact with many middle/high school math teachers and young university math educators.
My own experience on this is to
understand what Fuller is talking about you need to have done some of your own work; make
models in a variety of materials, have done 2-D exploration, find
understanding through your own observations about geometry in this world. Without that one has little context for understanding the words he uses. To
read Fuller’s work is not enough. His
language is not a problem if you have your own experience to bring, otherwise it gets passed over, reading as a passive state. If Fuller's
writing does not stimulate the reader to put the book down, roll up their selves and do
their own work, then their time is better spent reading something else. Fuller gives context for his own exploration,
demonstrates the necessity of doing ones own work and taking the responsibility
for the contribution we can each make if we are stimulated enough to take on
our own lives as a project that has value to humanity. It is not so much about what he left as how he lived his life that allowed him to leave what he did.
As for your visual style with the video, I am a little confused. I need to watch it again to get a better sense
of what you are doing. I know it is not without thought; I’m just a little unclear
on the first go around about how you have used the visual to inform my experience. Maybe
on the next viewing it will make sense. In the mean time I am curious to see
were you go with the next video.
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kirby urner
There's some stress in that I emphasize getting to the point quickly, not wasting viewers' time, but then I seem to ramble and fumble, not getting to said point right away.
Only about seven minutes.
https://youtu.be/aOajXg2_x_c
The core topic is "Quadrays" per the cute comic book cover that iMovie supplies right at the beginning.
I don't mind the hokey amateur hour ham radio aspect of what I'm doing, because it proves to viewers there's zero barrier to entry, just some awareness of the content needed. I set a low bar in dimensions that don't matter, but keep to the Synergetics pretty strictly, more than most have attempted (in terms of adhering to his vocabulary, down to the 4D part). Quadrays: a useful puzzle piece.
We typically use an corner of an average box (cardboard, maybe even metal) to make that point, showing edges of "height, width and depth" emanating therefrom, each at 90 degrees to the two others.
The educational authorities in the art and science of "plane" and "solid" geometry disregard the environmental otherness: They assume an infinitely extendible imaginary plane upon which they mark apart two infradimensional imaginary points A and B between which they can draw an imaginary shortest straight line whose "length" AB constitutes their academic mathematicians' first-dimensional state. They then mark apart on the same infinite imaginary plane third and fourth points C and D, which are then linearly interconnected by another "straight" line CD in the same imaginary plane with, parallel to, and at an AB distance from, line AB, with a third line CA drawn in the same plane perpendicular to line AB at A, and a fourth line DB in the same imaginary plane drawn perpendicular to line AB at B, whereby either of the lines CA or DB constitutes the "breadth," which is the educators' second-dimensional state. They then erect four AB __ long lines perpendicular to the first imaginary plane at points A', B', C' and D', respectively. They then draw the imaginary straight lines A'B', B'C', C'D', and D'A'. With all this so-called construction-which would collapse in the presence of gravitational reality __ they have now attained their third-dimensional state of "height" above their two- dimensional square plane base. This assumedly produces three-dimensional reality, which by virtue of their constructional strategy suggests to them that reality is only cubically measurable or comprehensible.
http://www.rwgrayprojects.com/synergetics/s05/figs/f27703.html
This makes him a rebel, an outlaw. That's his role, vis-a-vis the status quo -- in many dimensions.
The minimum family of inherent systemic omnicosmic interrelationships is inherently primitive and eternal. Primitive dimensionality is expressible only in terms of the interproportionality of the components of whole minimum systems__ergo, in prefrequency primitive tetravolume proportionality and the latter's primitive topological characteristics. There can be no partial systems. Systems can be divided multiplyingly only into whole systems.
Res Extensa is made of "dough" (big bang: "doh!") and until we add time (the only "real" dimension, wherein energy occurs) none of this "reality" really "exists" (it's prefrequency) -- but nevertheless "right out of the gate" (ab initio) it's 4D already. A tetrahedron is born. It has shape and therefore angles.
There's still "growing bigger" and "growing smaller" of course (like Alice) -- which is where the other term "frequency" comes in, and limits to one's ability to further subdivide in a given context (getting us to "the points" of that system e.g. its "grains of sand"). We have "angle and frequency" with which to talk about shape, time and size.
However if that's to be done at all deeply (at more than a superficial level), then we need to come to grips with this "questioning of the prevailing orthodoxies" vocabulary in the realm of elementary mathematics. We need to break it to the children that an authorized establishment genius may have shared their intuitive doubts about the "my way or the highway" attitude assumed by some Euclidean Geometry teachers (not all -- many well understand we have room for many sandcastles on this axiomatic beach).
Recasting the historical narrative to include Fuller's voice the job of the philosophy department, I argue. Sure, most faculty have chosen other projects for themselves, so the work has been slow, but I think we've just about finished laying the groundwork (just about nailed it) in the form of an accessible kid-friendly Martian Math (nice whole number volumes). Stay tuned and watch your TV Guide [tm].
Bradford Hansen-Smith
Kirby, you have pointed out point, line, plane, and cubic volume are only concepts. As with many things even this concept we have backwards. There is appeal for having constructed a logically simplistic system of linear progression, meaning simple for the inattentive mind to take in, simple for the teacher. Starting with spherical volume, a point without scale, it is either divided into or expanded outward to planer definition that is referenced by boundary edges that are defined by relationships between reference points in space; division creates multiples that can be added and subtracted. Four points to a tetrahedron does not deny 3-D space where the fourth dimension is movement giving relationships thus meaning to static generalizations. The center point of the tetrahedron is also center point to the octahedron, neither centers exists in spherical packing except in the VE where it has to be removed to allow movement of contraction and expansion. The tetrahedron is never without its dual; in the 4-D quadray system is radial expansion from center to triangle face, a two size tetrahedra compound of 8 points.
Words have different meaning depending on how we order them to our experience. Using them to a preordered placement to standards of the time they were used makes it difficult to re-concept and carry forward. Words are like pieces of clay that we stick together to form images in the mind, adding and subtracting to better articulate what starts out as lumpy ideas. When we get what we like we bake it until it is hard and unforgiving. There is a tendency to confuse ourselves by separating and oversimplifying. Clay and words are both materials, used to explore meaning of patterns, forms and systems of arrangements in a very highly ordered projected reality.
Any study of structural pattern shows the tetrahedron as primary twin, first born, origin to and in that regard primitive to all polyhedral systems. The tetrahedron is never singular except as a formed concept in isolation, otherwise it is always two or more. I think Fuller made that clear. The cube is a two-tetrahedra compound symmetrically balanced that when revealed in the isotropic vector matrix is an unformed relationship. (Reference Sept 10th paper on stick weaving for dual tetra-cube transformation.) The real value of cubic measure is that it has brought us to realize this process of pattern formation is not about cubes nor is it about measurement, rather proportional movement that measuring keeps track of. The cube is only structural when understanding tetrahedral dual nature.
Where I disagree with Fuller, and this may be a generational word thing, is that there are no whole systems. Systems are always plural, little systems within larger systems spread out in space through time. The only system singular whole is the inclusive nature of whole itself. Whole is unity, complete, healthy to perfection, in alignment to all we have ever known, can ever know, and all that is far beyond our capacity as creatures less than perfection, can ever know. Whole is unity absolute, source to all possible systems. I think Fuller understood this. The “Whole systems” concept must have been a stretch for his generation, and still is for we continue to isolate, fragment and cut things apart where we find coherent parts we call whole systems.
I agree with you, Fuller’s explanations are often
a bit difficult to follow in the abstract since they express his experience
that came from his own work using words of his own making and arranging. His
philosophy came directly from the work he did which is why it did not aligne
to any philosophical pedagogy where people think without the use of their
hands. He had an interesting capacity to find great meaning from the work he
chose to do through his own observation and imagination relating to
hands-on touching humanity to universe. The word universe now has become plural
in less than twenty-five years as our understanding increases about where we
are and the proportional interrelationships between systems of one order. My
sense is we are reading different pages in the same book; good story.
kirby urner
Polyhedrons are everywhere! "Aristotle was right (about tetrahedrons filling space), remember the MITE (AAB+ or AAB-)." That's a little campaign I like to run periodically (in a PR sense). There's this ancient debate in the literature wherein Aristotle's detractors take him to task for his erroneous assertion that tetrahedrons fill space. His apologist-defenders (me one of them in this episode) counter the translation leave it open he meant irregular and even in the 1900s we were still learning about what identical irregular tetrahedrons fill space. Sommerville is a name to look up. He and Fuller overlapped in their inventories (Fuller missed the 1/4 RITE as a nameable Sommerville) and Aristotle may be cited appreciatively for helping to get the ball rolling.
kirby urner
Polyhedrons are everywhere! "Aristotle was right (about tetrahedrons filling space), remember the MITE (AAB+ or AAB-)." That's a little campaign I like to run periodically (in a PR sense). There's this ancient debate in the literature wherein Aristotle's detractors take him to task for his erroneous assertion that tetrahedrons fill space. His apologist-defenders (me one of them in this episode) counter the translation leave it open he meant irregular and even in the 1900s we were still learning about what identical irregular tetrahedrons fill space. Sommerville is a name to look up. He and Fuller overlapped in their inventories (Fuller missed the 1/4 RITE as a nameable Sommerville) and Aristotle may be cited appreciatively for helping to get the ball rolling.
At least superficially.
The caveat has to do with the Synergetics dissection of the space-filling MITE into (A+ A- B-) and (B+ A+ A-) triplets of A & B modules, where one has two As of opposite handedness, plus a B on either side. Either choice fits the same MITE mold i.e. from the outside it's not evident what Bias (B for Bias) applies.
A and B stem from dissections of the home base Tetrahedron (volume 1) and the cube-dual Octahedron (volume 4) respectively. The dissections are pretty obvious and both A and B have the same volume of 1/24.
Ergo the MITE weighs in at 1/8 in the pre-frequency (4D) account (i.e. we have yet to add enough energetic dimensions (frequency**3) to assign a "landscape position" for our construction, with regard to an actual planet or anything; it's not a sculpture made of any particular materials (yet)).
kirby urner
Finally, absolutely about the Tetrahedron being a Twin phenomenon, Stella Octangula, the Cube's innards. We've got the bridge concept of Duality to unify the Platonic five as three pairs, so why only five and not six then? Because of course the tetrahedron is self-dual. The other two pairs: icosahedron - dodecahedron; cube - octahedron. We could start their, a good Genesis point (from which to grow out the rest).
Bradford Hansen-Smith
Kirby, why 5 and not 6 is a question for anyone that thinks about the Platonic solids. They are both right, same as octahedron is both regular and truncated at the same time. We begin to see everything as multi functional parts interrelated through circle/point origin. Three sets of two is a 3-6-12 symmetry not 5-10. The 4 points plus 6 edge relationships is 10. I don’t count faces since they are a function of the form and not pattern. The faces are important when looking at properties which for the tetrahedron are 4+6+4=14 (1+4=5). Can’t get away from two 5’s moving to next frequency level 10; the first fold of circle in half. The properties of the tetrahedron compact down to 5. The properties for the other four regulars, a pair of duels, compact down to 5. All five regular polyhedra together are number 10. The three regular sets of duels can be formed by folding and joining multiple circles. The circle is where I start and end; always a circle regardless of how it is reconfigured or in what number of combinations.
Fold 5 tetrahedra. Stellate each face of one with the other 4 showing the tetra dual with 5 circles. Next fold 10 circles into 10 tetrahedra. Open and join 2 forming the octahedron leaving 8, (2 sets of 4.) Next stellate alternate faces of the octahedron attaching 4 tetrahedra. Beautiful; a solid 2-frequency tetrahedron. Then join remaining tetrahedra on remaining 4 open faces of octahedron to get the Stella Octangula. One of many such circle folding exercises for primary grade students. For some it is difficult to see the cube, so they string tape between the 8 points making it easy to see the cubic relationship of tetrahedra. The problem with defining the cube as solid and reciting its properties is we miss it is a relationship of tetrahedra.
The same process goes for making the icosahedron. Using 4 open tetrahedra, 4 triangles each, make an icosahedron. There is both right hand and left hand arrangement in making an icosahedron from 16 triangles, not seen in traditional approach.There are 20 faces where 4 are open in a tetrahedron pattern. 20 more tetrahedra are used for three
stages of stellateing the icosahedron; the tetrahedron with an icosahedron
center, the cube with an icosahedron center, and the dodecahedron with an
icosahedron center, using 24 tetrahedron formed circles. Again by stringing the
20 points we see the 12 open pentagon planes. This gives primacy to the three
regular polyhedra formed from triangle faces. This does not stay within the
boundaries of traditional classification nor does it utilize current
technology, but students understand as they discuss what they have done, and
they will never forget it.
Volume is a way to compare the structural pattern of regular polyhedra where the tetrahedron is not only obvious as a primary unit of measure but is consistent that all tetrahedra are congruent to the same measure of circle where everything is structurally formed. It could be imagined this might be how Martians introduce Math to their young.
Mites are interesting little tetrahedra, but I do not see them as important "building blocks," only as divisional sub-units of the tetrahedra. Years ago I divided the tetrahedron into 24
of them and then divided them into 48 units hinging them together in a torus
ring necklace that reformed to the tetrahedron; didn’t want to lose any of
them. Can’t give you any pictures, I have subsequently misplaced them in storage unit, which is like losing them.
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kirby urner
Kirby, why 5 and not 6 is a question for anyone that thinks about the Platonic solids. They are both right, same as octahedron is both regular and truncated at the same time.
Two columns of three makes it look more like six total, but the Tet and Dual Tet are both Tets.
Icosahedron (edges D) + Dual Icosa = RT (vol: 120 * E)
We begin to see everything as multi functional parts interrelated through circle/point origin. Three sets of two is a 3-6-12 symmetry not 5-10.
http://www.rwgrayprojects.com/synergetics/s09/figs/f87137b.html
The 4 points plus 6 edge relationships is 10. I don’t count faces since they are a function of the form and not pattern.
We may imagine we can imagine "no volume" i.e. "the universe before the big bang" but as soon as there's a flicker of light (i.e. fire) there's your tetrahedron in view.
The faces are important when looking at properties which for the tetrahedron are 4+6+4=14 (1+4=5). Can’t get away from two 5’s moving to next frequency level 10; the first fold of circle in half. The properties of the tetrahedron compact down to 5. The properties for the other four regulars, a pair of duels, compact down to 5. All five regular polyhedra together are number 10. The three regular sets of duels can be formed by folding and joining multiple circles. The circle is where I start and end; always a circle regardless of how it is reconfigured or in what number of combinations.
Here at the construction site, we pour bucket loads of concrete from either side (Martian or Earthling) to build the dam itself.
The Martians have this tetrahedron of edges D they consider unit, whereas we-the-Earthlings have our "3D" cube of edges R, that's our "home base" unit -- for some reason most Martian's can't fathom, which is where the ambassadors come in, to 'splain it to 'em. For one thing, cubes don't tip over as easily, if used as mugs (laughter -- an ice breaker).
Fold 5 tetrahedra. Stellate each face of one with the other 4 showing the tetra dual with 5 circles. Next fold 10 circles into 10 tetrahedra. Open and join 2 forming the octahedron leaving 8, (2 sets of 4.) Next stellate alternate faces of the octahedron attaching 4 tetrahedra. Beautiful; a solid 2-frequency tetrahedron. Then join remaining tetrahedra on remaining 4 open faces of octahedron to get the Stella Octangula. One of many such circle folding exercises for primary grade students.
I'm teaching adult philosophy students mostly. We study Hegel, Heidegger, Fuller and Wittgenstein with a little Kierkegaard and Kaufmann around the edges. From these greats, one may branch to others.
My friend Applewhite (collaborator on Synergetics) liked the Fuller + Wittgenstein combo quite a bit. That's something I've developed. Quadrays are a "language game" we use to investigate the foundations of mathematics.
Is Kantian space 3D or 4D? Verdict: it's cultural.
We also do some mathematics of course (philosophers always have), writing in Python perhaps (very pithy). We might watch some of your Youtubes and have some paper plate sculptures in the Gallery.
For some it is difficult to see the cube, so they string tape between the 8 points making it easy to see the cubic relationship of tetrahedra. The problem with defining the cube as solid and reciting its properties is we miss it is a relationship of tetrahedra.
The same process goes for making the icosahedron. Using 4 open tetrahedra, 4 triangles each, make an icosahedron. There is both right hand and left hand arrangement in making an icosahedron from 16 triangles, not seen in traditional approach.There are 20 faces where 4 are open in a tetrahedron pattern. 20 more tetrahedra are used for three stages of stellateing the icosahedron; the tetrahedron with an icosahedron center, the cube with an icosahedron center, and the dodecahedron with an icosahedron center, using 24 tetrahedron formed circles. Again by stringing the 20 points we see the 12 open pentagon planes. This gives primacy to the three regular polyhedra formed from triangle faces. This does not stay within the boundaries of traditional classification nor does it utilize current technology, but students understand as they discuss what they have done, and they will never forget it.
Volume is a way to compare the structural pattern of regular polyhedra where the tetrahedron is not only obvious as a primary unit of measure but is consistent that all tetrahedra are congruent to the same measure of circle where everything is structurally formed. It could be imagined this might be how Martians introduce Math to their young.
We'll infuse some Martian thinking, that's unavoidable, but there's no "takeover" planned. Humans are known to be violent and paranoid. Partly why the Martians left their kids home (see "ratings" discussion) and surrounded their apartment with a LuxBlox fence [tm] is they know humans dislike anyone defying their Orthodoxies. "When in Rome..."
So we needed a quick intro that's over quick, and Martian Math is it. Takes maybe 10-15 minutes to get it in brief, with more background optional, and you're good to go. Next time someone ways something knowing about Fuller's Synergetics, you'll know to nod knowingly as you'll be in on it too. Did you know there's a Synergetics Dictionary in four volumes (Garland Press) now on-line?
Start with the Platonic Five, dual combine, rescale, spin, subdivide, and you're done (all in 4D tetravolumes). Remember about special cases (versus generalized principles) but that's in other authors already i.e. most of Synergetics is not all that new, just freshly recast in the new-style prose of American Transcendentalism. Angle vs Frequency.
Mites are interesting little tetrahedra, but I do not see them as important "building blocks," only as divisional sub-units of the tetrahedra. Years ago I divided the tetrahedron into 24 of them and then divided them into 48 units hinging them together in a torus ring necklace that reformed to the tetrahedron; didn’t want to lose any of them. Can’t give you any pictures, I have subsequently misplaced them in storage unit, which is like losing them.
Cubes and Rhombic Dodecahedrons analyze nicely into MITEs and from these dissections we develop the Coupler, the 8-MITE oblate Octahedron that's also a space-filler.
http://www.newciv.org/Synergetic_Geometry/charhawk/mods.html
Those into Archimedian geometry (dual space-fillers) are very familiar with this shape, shown on this diagram (chart) by Guy Inchbald for example:
http://www.steelpillow.com/polyhedra/AHD/AHD_Chart.pdf
kirby urner
On Mon, Nov 9, 2015 at 6:36 AM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:Kirby, why 5 and not 6 is a question for anyone that thinks about the Platonic solids. They are both right, same as octahedron is both regular and truncated at the same time.
It would be easy to present the Platonic Five (as the set is usually called) with Duals aligned across from each other (for example):Tet <--> Dual TetOctahedron <--> Dual Octa (i.e. Cube)Icosahedron <--> Dual Icosa (i.e. Pentagonal Dodecahedron or PD)keeping those with an all-triangle structure down the left, with the remainder derived by "dualizing" or whatever we call that operation.
Two columns of three makes it look more like six total, but the Tet and Dual Tet are both Tets.In the concentric hierearchy of Synergetics, we start to sprinkle in the relative volume numbers as well as combining duals (+):
Icosahedron (edges D) + Dual Icosa = RT (vol: 120 * E)
Actually I made a notational error here.
I was looking it up in Python source code and inadvertently turned E3 to E:
The original reads:
print("SuperRT: {:10.7f}".format(120 * E3)) # ромбические триаконтаэдра
Capital E means "scaling up" (getting bigger) whereas lowercase e means "scaling down" by φ.
When I shrink all of E's edges by Phi, it becomes e3. When I expand all its edges by φ, it becomes E3.
s3 = Svol * pow(φ, -3)
s6 = Svol * pow(φ, -6)
e3 = Evol * pow(φ, -3)
E3 = Evol * pow(φ, 3)
φ = sqrt(5)/2 + 0.5
D = 1.0
R = D/2
# Fig. 986.411A T & E Module
# http://www.rwgrayprojects.com/synergetics/s09/figs/f86411a.html
h = R
OC = h
OA = h * sqrt((5 - sqrt(5))/2)
OB = h * sqrt((9 - 3 * sqrt(5))/2 )
CA = (h/2) * (sqrt(5) - 1)
AB = h * sqrt(5 - 2 * sqrt(5))
BC = (h/2) * (3 - sqrt(5))
Emod = ПлейнНэт(OC, OA, OB, CA, AB, BC)
Evol = объем(Emod)
print("Emod volume = :", Evol)
# Fig. 986.411A T & E Module
# http://www.rwgrayprojects.com/synergetics/s09/figs/f86411a.html
h = R * pow(2/3,1/3) * (φ / sqrt(2))
OC = h
OA = h * sqrt((5 - sqrt(5))/2)
OB = h * sqrt((9 - 3 * sqrt(5))/2 )
CA = (h/2) * (sqrt(5) - 1)
AB = h * sqrt(5 - 2 * sqrt(5))
BC = (h/2) * (3 - sqrt(5))
Tmod = ПлейнНэт(OC, OA, OB, CA, AB, BC)
Tvol = объем(Tmod)
print("Tmod volume = :", Tvol)
Bradford Hansen-Smith
Kirby, as you have indicate we work on different levels with the same geometry, which I have for my own clarification comprehensively renamed Wholemovement, I have done this not from disrespect or discard, but to better understand the breadth and depth of the discipline as we move forward pushing limitations from the past. I think you might have come up with Martin Math for similar reasons.
I apologize for not being clear. When I refer to the circle, it is not to the “great circle” abstraction of traditional geometry, I understand that. Compressing a sphere does not give a great circle; it shows five congruent circles. Traditionally defined as a disk, very thin cylinder, not tracing around the sphere or cutting in half. When folded the circle forms two opposing sets of two tetrahedra, not obvious without observing two imaginary points that generate two more points at creased ends (4 points, 4 tetrahedra.) The difference between drawing a line between two points and touching points to get a line half way between at right angle to the distance between points is the difference between 2-D and 3-D as lined out at 90 degrees. There must be some theorem that addresses this as bases for the hypercube and extrapolating higher dimensions.
I was not clear in stating 3-6-12: referring to the sequential development of folds from the circle in half. As you point out this sequence is also observed in the properties of the cube, but that was not my context. That 3 diameters shows 6 areas and 6 diameters shows 12 areas as a function of 6 folds in a ratio of 1:2,. Where 4 diameters show 8 areas, and 5 diameters show 10 areas are the only other two symmetries that can be folded from that first fold in half. All folds are right angle folds, which gives 90 degrees precedence, but not in cubic form; echoed in Martian Math. Since number sequences are generalizations I often forget to give a context, thinking the is commonly understood.
Initially to understand geometry and math I have had to approach it from a child’s perspective, questioning where things come from, tracing back to origin, otherwise I did not and still lack understanding about the plucked abstractions of math or the patterns that order the forms of geometry. This is still an open question. Possibly this is why I incorporated young children in my exploration early on and still consider them as primary focus for what I do. When you refer to the abstract math formulation and computer programs, such as Python and other semiotic systems, I am lost, not being familiarity with or conversant in those languages. Maybe some day that will change, but for now I get more from the hands-on experience, even with all the possibilities in a virtually abstracted world. I think you have limited experience with folding circles, but do assume you have at least folded Fuller’s spherical VE using four circles. We share words and symbols but in different experiential context. for me that makes our discussion challenging and interesting.
When I refer to the tetrahedron vertex points you give context with the face center points of radial locations from axial center with “atomic integrity.” Both simultaneously are different levels of true. Together they are the dual tetrahedra, each a different scale, as are the two sets of two tetrahedra from the first fold in half that are different sizes. I try to keep it simple and observable for my childlike need-to-see-it-touch-it understanding. I enjoy your experience and indulgence in imagination and tangential fantasies. How far is science of the imagination from science of the observable skin of life? It seems geometry is a generalization of what we can see and imagine it to be, but is not that at all. Maybe your right, the bedevilment is in the details that keeps us from the greater truth, but engages us in the experience and seeking beauty in how it all fits together getting us closer to a truer realization about the whys, whats, wheres, and how comes.
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