60 research outputs found
On the Dissection of Rectangles into Right-Angled Isosceles Triangles
AbstractWe consider the problem of dissecting a rectangle or a square into unequal right-angled isosceles triangles. This is regarded as a generalization of the well-known and much-solved problem of dissecting such figures into unequal squares. There is an analogous “electrical” theory but it is based on digraphs instead of graphs and has an appropriate modification of Kirchhoff's first law. The operation of reversing all edges in the digraph is found to be of great help in the construction of “perfect” dissected squares
An enumeration of equilateral triangle dissections
We enumerate all dissections of an equilateral triangle into smaller
equilateral triangles up to size 20, where each triangle has integer side
lengths. A perfect dissection has no two triangles of the same side, counting
up- and down-oriented triangles as different. We computationally prove W. T.
Tutte's conjecture that the smallest perfect dissection has size 15 and we find
all perfect dissections up to size 20.Comment: Final version sent to journal
Tilings of an Isosceles Triangle
An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union
of N trianglescongruent to T, overlapping only at their boundaries. The
triangle T is the "tile". The tile may or may not be similar to ABC. In this
paper we study the case of isosceles (but not equilateral) ABC. We study three
possible forms of the tile: right-angled, or with one angle double another, or
with a 120 degree angle. In the case of a right-angled tile, we give a complete
characterization of the tilings, for N even, but leave open whether N can be
odd. In the latter two cases we prove the ratios of the sides of the tile are
rational, and give a necessary condition for the existence of an N-tiling. For
the case when the tile has one angle double another, we prove N cannot be prime
or twice a prime.Comment: 34 pages, 18 figures. This version supplies corrections and
simplification
Hinged Dissections Exist
We prove that any finite collection of polygons of equal area has a common
hinged dissection. That is, for any such collection of polygons there exists a
chain of polygons hinged at vertices that can be folded in the plane
continuously without self-intersection to form any polygon in the collection.
This result settles the open problem about the existence of hinged dissections
between pairs of polygons that goes back implicitly to 1864 and has been
studied extensively in the past ten years. Our result generalizes and indeed
builds upon the result from 1814 that polygons have common dissections (without
hinges). We also extend our common dissection result to edge-hinged dissections
of solid 3D polyhedra that have a common (unhinged) dissection, as determined
by Dehn's 1900 solution to Hilbert's Third Problem. Our proofs are
constructive, giving explicit algorithms in all cases. For a constant number of
planar polygons, both the number of pieces and running time required by our
construction are pseudopolynomial. This bound is the best possible, even for
unhinged dissections. Hinged dissections have possible applications to
reconfigurable robotics, programmable matter, and nanomanufacturing.Comment: 22 pages, 14 figure
Stereometry activities with DALEST
This book reports on a project to devise and test a teaching programme in 3D geometry for middle school students based on the needs, knowledge and experiences of a range of countries within the European Union. The main objective of the project was the development (and testing) of a dynamic three-dimensional geometry microworld that enabled the students to construct, observe and manipulate geometrical figures in space and which their teachers used to help their students construct an understanding of stereometr
Compound Perfect Squared Squares of the Order Twenties
P. J. Federico used the term low-order for perfect squared squares with at
most 28 squares in their dissection. In 2010 low-order compound perfect squared
squares (CPSSs) were completely enumerated. Up to symmetries of the square and
its squared subrectangles there are 208 low-order CPSSs in orders 24 to 28. In
2012 the CPSSs of order 29 were completely enumerated, giving a total of 620
CPSSs up to order 29.Comment: 44 pages, 10 figures. For associated pdf illustrations of enumerated
compound perfect squared squares up to order 29, see
http://squaring.net/downloads/downloads.html#cps
Euclid’s Elements for High School Classrooms
What is our goal when teaching students geometry? Is it to obtain a passing grade, be able to construct geometric figures, know all of the necessary terms? I would like to propose that the purpose is to cultivate a love for the logic, art, and argument of geometry. The original geometry book written by Euclid was used as a guide for two thousand years and led many of our historical geometers to discoveries. Instead of relying on memorizing confusing acronyms for congruent figures, let\u27s give the students an opportunity to see the work behind the properties. By using the methods outlined in this curriculum, students will experience geometry, study how to accurately justify their work, learn how to discuss, disagree, and defend respectfully, and know the math behind the theorems. To evaluate the curriculum, a small pilot study was conducted on the impact on student learning. Moreover, the impact and value of this curriculum was evaluated by two experts who specialize in teaching mathematics at the high school level
The Essence of Mathematics Through Elementary Problems
"It is increasingly clear that the shapes of reality – whether of the natural world, or of the built environment – are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself ‘tick’, and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside. The Essence of Mathematics consists of a sequence of 270 problems – with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline. The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works.
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