Classicaldifferentialgeometry curvesandsurfacesineuclideanspace. Gardner 7 was the first to state cutting out com plex polygons as an open problem. We are given an initial shape drawn on a piece of paper. I have been impatiently waiting to try this activity with others. Fold it in half and you can do it in two turns, no jabbing required. Circuit analysis superposition thevenins and norton. This is fold and cut theorem numberphile by mr tomkins on vimeo, the home for high quality videos and the people who love them. Mar 23, 2016 in this short video clip, dr katie steckles gives a very brief explanation of the fold and cut theorem. Freedman chair au, thomas kwokkeung, approximating ehomotopy equivalences by homeomoephisms on 4manifolds 1990, michael h. It can, at times, be a little fiddly and time consuming because, depending on the level of detail in a pattern, you may have to deal with very many folds per page alot of which may be of just a mm in size. The author thanks tibor jord an for calling our attention to the work 1 by b ohme.
Because each angle is 35, then we can state that a ll b. Below we will give some examples of using pascals theorem in geometry problems. These theorems are also applicable to a certain higher order ode since a higher order ode can be reduced to a system of rst order ode. Adventures in cutting paper today well explore a couple cool areas of math which feature cutting paper in interesting ways. Superposition, thevenin norton equivalent circuits. Demaine is and the fold and cut theorem debrief mindset message 5 min remind students of the video messages they heard that there is no such thing as a math brain or a math person. An incomplete guide to its use and abuse, is for the general reader. It is inspired by the fold andcut theorem, in particular the demainedemainehawskleyitolohmanberstephens 2010 paper about simple fold and cut. The fold andcut theorem states that any shape with straight sides can be cut from a single idealized sheet of paper by folding it flat and making a single straight complete cut. Can you cut a square from the center of a piece of paper with just one straight cut. In this numberphile video, manchesterbased mathematician dr. Consider the following example of a linear circuit with two sources.
Make a square within a square and prove pythagoras theorem take a square sheet of paper. If you are given a figure see below with congruent corresponding angles then the two lines cut by the transversal are parallel. The fold and cut theorem says that any collection of straight lines can be cut in this way, that is if you have a piece of paper which can be folded any number of times and scissors which can cut through any number of layers of paper. If three sides of one triangle are congruent to three sides of a second triangle, then. In light of these, the goal of our present quick proof is that this perhaps not so wellknown proof is now available in a short and more or less selfcontained form. Then well start to explore the fold and cut theorem developed by erik. Jul 11, 20 the dead puppy theorem and its corollaries 26 letters to a first year teacher. Fold and cut theorem numberphile safe videos for kids. If x,t is a regular space with a countable basis for the topology, then x is homeomorphic to a subspace of the metric space r the way i stated the above theorem, it is ambiguous. One application of the foldandcut theorem is the design of the logo for cccg 2001.
The structure theorem for the cut locus of a certain class of cylinders of revolution ii article pdf available in tokyo journal of mathematics 372 december 20 with 23 reads. In the limit, a and b will coincide and the line ab will become the tangent line at b. Nortons theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load. Now, let pand qbe recursive relations such that for a2. Find the thevenin equivalent circuit of the circuit. Such shapes include polygons, which may be concave, shapes with holes, and collections of such shapes i.
Visual connect in teaching in the classroom paper folding and. Some applications of the cutelimination are considered. Let fx and fy denote the partial derivatives of f with respect to x and y respectively. The earliest known description of the so called fold and cut problem comes from japan in 1721. The origami polygon cutting theorem the origami polygon cutting theorem tries to solve the fold and cut problem. Its in uence on the law has been especially profound. Just as with thevenins theorem, the qualification of linear is identical to that found in the superposition theorem. Agol, ian, topology of hyperbolic 3manifolds 1998, michael h.
The theorem is that every pattern plane graph of straightline cuts can be made by folding and one complete straight cut. Joseph orourke briefly describes two foldandonecut methods in his computational geometry column 36, in the international journal of computational geometry and. Ramanujan applied this theorem to compute several definite. The technical report computing extreme origami bases studies the special case of cutting out a convex polygon. Sep 25, 2015 katie steckles discusses the fold and cut theorem from a to z. In this short video clip, dr katie steckles gives a very brief explanation of the fold and cut theorem.
Thus it is possible to make single polygons possibly nonconvex, multiple disjoint polygons, nested polygons, adjoining polygons, and even floating line segments and points. Make your own font, 1 cut at a time scientific american. From this perspective the implicit function theorem is a relevant general result. There are two ways to find the folds needed to do this. The angle bisector theorem stewarts theorem cevas theorem solutions 1 1 for the medians, az zb. A convex polygon is simple fold and cuttable if and only if it has a line of re ectional symmetry.
The fold and cut theorem states that any shape with straight sides can be cut from a single idealized sheet of paper by folding it flat and making a single straight complete cut. Fold andcut theorem claims the possibility to cut out from a sheet a set of straightline drawing using only one cut of scissors, without producing any other cut in the sheet and separating all the figures at the same time, just by folding flat the paper before the last cut. P ostulates, theorems, and corollaries r2 postulates, theorems, and corollaries theorem 2. In any sufficiently strong formal system there are. We remark that there are limiting cases of pascals theorem. Making polygons by simple folds and one straight cut. Some of our earlier papers on the foldandcut problem are the following. If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. Rolles theorem, mean value theorem the reader must be familiar with the classical maxima and minima problems from calculus. Just as with thevenins theorem, the qualification of linear is identical to that found in. Circuit analysis superposition thevenins and norton theorem. Theoremsabouttriangles mishalavrov armlpractice121520. Fold the bottom right corner towards the diagonal, so that the edge of the sheet lies parallel to the diagonal. In most cases the lower bound is not very good, in the sense that the interval on which the solution exists may be much larger then the interval predicted by the theorem.
Whilst it may look difficult and complicated, cut and fold book folding is actually very easy to do. More information on the history of this theorem can be found in 5 p. Ramanujans master theorem states that, under suitable conditions, the mellin. The dead puppy theorem and its corollaries 26 letters to a first year teacher. Cutting a square off the center of a paper is easy. A convex polygon is simplefoldandcuttable if and only if it has a line of re ectional symmetry. Visual connect in teaching in the classroom paper folding. Katie steckles discusses the fold and cut theorem from a to z. The origami polygon cutting theorem the origami polygon cutting theorem tries to solve the foldandcut problem. Demaine and alii, allows a wide exploration of mathematical issues. Externalities and the \coase theorem the \coase theorem has been one of the most in uential contributions to come from economics in the last fty years. You can cut out any shape you want with one cut if youre willing to do some clever folding.
Katie steckles demonstrates an idea that dates back to 1721 in japan. The origami polygon cutting theorem mcgill university. Sep 23, 2015 the most amazing thing about this is that theres a theorem in mathematics about this which says, as long as a shape is made up of straight lines, there is always a way to fold it properly such that you get that shape with a single straight cut. Did you know that you can cut any shape made of all flat edges by folding a single piece of paper and cutting once. Some applications of the cut elimination are considered. Cevas theorem the three lines containing the vertices a, b, and c of abc and intersecting opposite sides at points l, m, and n, respectively, are concurrent if and only if m l n b c a p an bl cm 1 nb malc 21sept2011 ma 341 001 2. Gave proofs of the theorems and conjectures of ramanujan obtained by him through. As long as you avoid curvy letters, you can do this for every letter in the english alphabet. The existence and uniqueness theorem are also valid for certain system of rst order equations. Mathematics awareness month april 2014 fold and one cut. The fold and cut theorem states states that any shape with straight sides can be cut from a single sheet of paper by folding it flat, possibly with many folds, and making a single straight complete cut.
If two lines are cut by a transversal so that the exterior angles on the same side of the transversal are supplementary, then these lines are parallel. Theorem the theorem is that every pattern plane graph of straightline cuts can be made by folding and one complete straight cut. Anyone can learn any level of math with hard work and effort. Why dont you have a go at folding a piece of paper and cutting out a triangle with just one straight cut, as katie suggests. With a resistor, while nortons theorem replaces the linear circuit with a.
The foldandcut theorem states states that any shape with straight sides can be cut from a single sheet of paper by folding it flat, possibly with many folds, and making a single straight complete cut. This clip accompanies our other videos in which katie explores some mathematics of folding. Katie got into this fold and cut problem by trying to cut a square out of a piece of paper as efficiently as possible. If two lines are cut by a transversal so that the corresponding angles are congruent, then these lines are parallel. Among these, we prove that basic logic has the disjunction property dp and the explicit. This typeface illustrates how to make each letter and numeral in the alphabet by taking a square piece of paper, making a sequence of simple folds each folding along one line, and cutting along one complete straight line. Ramanujan theorems pdf the history of the notebooks, in brief, is the following. U rbe a smooth function on an open subset u in the plane r2. Therefore, if kuhnscomment is correct and i believe it is, then the reader ought to conclude. Whenever i say this, the unfinished sentence in my head is fold and cut, baby. The theorem asserts that any shape with straight sides can be cut from a single idealized sheet of paper by folding it flat and making a single straight complete cut. First well take a look at cylinders and m obius strips and generalizations with more halftwists.
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