A Math Forum Project


Geometry Project of the Month

Rep-tiles - March 1996

  1. Can you find a way to divide any triangle into 4 congruent similar triangles?

  2. How would you divide squares, rectangles, and parallelograms?

  3. Can you find other quadrilaterals that are rep-tiles? There is at least one kind of trapezoid (which has an neat subdivision).

  4. How about other polygons? Are there things that make a polygon a good candidate for being a rep-tile?

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Submissions

From: visitor1@sasd.k12.pa.us
Amy Saveikis, Renee Tappe, Maria Briski
Grade 10
School: Shaler Area High School, Pittsburgh, PA

The general problem to be solved for this month's project is to 
explore different types of rep-tiles, or repetition tiles, for 
different types of polygons.

Rep-tiles for quadrilaterals like squares, rectangles, kites, and 
rhombuses can be found by:

  1. Taking the midpoints of each of the sides.
     Equation for midpoints: (x1 + x2), (y1 + y2)  
                                 2

  2. Connect the midpoints of the opposite sides.
     These steps will result in the formation of smaller congruent 
     quadrilaterals inside of the original quadrilateral.

The steps for dividing a triangle into four congruent smaller 
triangles the rep-tiles of any triangle are similar to the  
steps to finding the rep-tiles for a quadrilateral.

  1. Again find the midpoint of each side of the triangle.

  2. Connect consecutive midpoints of the triangle.

Connecting the consecutive midpoints of the sides will divide 
any triangle into four congruent smaller triangles.                                                                                 

Some qualities that make polygons good rep-tiles are if they 
are convex and all sides are equal.

In addition to this: Do rep-tiles have to be a smaller version of 
a larger shape or could a rep-tile be like the hexagon and 
pentagon pattern on a soccer ball or like the linoleum tile 
pattern of hexagons and diamonds. These patterns are composed of 
smaller polygons creating a larger whole, but are not smaller 
congruent parts of it.        


From: ruth@mathforum.org (Ruth Carver) Colleen Cusick & Joan Vivadelli Grade 10 School: Mount Saint Joseph Academy This week's project of the month was to answer questions and pose some of our own about "rep-tiles", or tiles that can be subdivided into a finite number of congruent tiles, each similar to the original tile. Here are our results: 1) You can divide any triangle into 4 congruent similar triangles by connecting the midpoints of adjacent sides. This makes three triangles with a vertex angle of the original triangle and one triangle formed by the lines connecting the midpoints. 2) A square would be divided by connecting the midpoints of opposite sides, forming four congruent triangles. A rectangle could be divided by connecting the midpoints of the legnths. Then you take the midpoints of the new rectangle formed and you get four congruent rectangles (the original rectangle has three lines perpendicular to and connecting the lengths with equal distances between them). A parallelogram can be divided in the same way as a rectangle except that the lines drawn to the lengths are not perpendicular to the lengths. 3) I found that another quadrilateral that is a "rep-tile" is the rhombus. A rhombus can be divided in the same way that a square is divided: the midpoints of opposite sides are connected. We could not find the trapezoid with the neat subdivision even though you hinted at it. 4) We think that a fewer number of sides is a characteristic of a "rep-tile". It is easier to connect midpoints and sides when you have fewer to connect. We feel that trangles and quadrilaterals are the easiest to divide. Another polygon that is a "rep-tile" is a pentagon. By forming a new pentagon around the vertex angles of the original one (and drawing sides from the midpoints of adjacent sides and then connecting those sides) you frorm five pentagons around the vertex angles and one in the middle, for a total of six. We were considering the option of having "rep-hedrons", being able to subdivided polyhedrons. We feel that the platonic solids would definately be "rep-hedrons"
From: ruth@mathforum.org (Ruth Carver) Cindy Spering Grade 9 School: Mount St. Joseph Academy 1) It is possible to divide any triangle into 4 congruent similar triangles. To achieve this, you must first find the midpoint of each of the sides. Then, connect those midpoints, with all of your lines to be drawn in the interior of the triangle. The result is 4 congruent, similar triangles. All of these triangles are similar to the original triangle. 2) Squares, rectangles, and parallelograms are easily divided. As long as the lines drawn are parallel to the sides of the figure, and evenly spaced, then the result will be a divided rep-tile into the desired amount of sections.Of course, there must be two sets of lines drawn, each set corresponding to the different set of parallels, (there are only two in these shapes). 3) Many quadrilaterals turn out to be rep-tiles. These include the: kite, rhombus, isoceles trapezoid. 4) Many-sided polygons do not turn out to be rep-tiles. This is because the shapes will not "interlock" with one another, at least with no gaps. The lesser the amount of sides and vertices helps to make a polygon a rep-tile. As you have seen, 3- and 4-sided polygons work well, but only if they have at least 2 congruent sides.(which, consequently, makes them also have a pair or more of congruent angles...). A regular hexagon does not produce itself - a nice flower shape is produced, though :^). The same holds true for all regular octagons, thus supporting the aforementioned.
From: ssusd2@owens.ridgecrest.ca.us Cassie Gorish Grade: 8 School: Murray Junior High 1. By connecting the midpoints of the sides of the triangle, you can make 4 congruent triangles. 2. Squares break down into squares, rectangles break down into rectangles, and parallelograms break down into parallelograms. 3. A perfect trapezoid (one with parallel top and bottom and equal sides) can break down into smaller perfect trapezoids. Rhomboids can break down into rhomboids.
From: anne_sandler@mathforum.org Kelly VanHusen and Susie Sandstede Grade: 9 School: Smoky Hill High School To divide triangles into smaller similar triangles in an equilateral and an isosceles triangle, you just draw a triangle in the center of the large triangle, which divides it into four similar triangles. In a right triangle, you draw a median to the hypotenuse, which forms two congruent triangles, and then draw medians in each of those to divide them into two congruent triangles each. The same thing in an obtuse triangle. You divide squares, rectangles, and parallelograms by drawing diagonals parallel to the sides of the figure. Regular trapezoids and kites are not rep-tiles. Isosceles trapezoids are - you divide those into right or equilateral triangles, and you can also divide the isosceles trapezoid into isosceles triangles. We found that figures that are rep-tiles had at least two things congruent, such as base angles, legs, or sides.
From: anne_d._sandler@shhs1.ccsd.k12.co.us Brian Christenson Grade: 9 School: Smoky Hill High School To divide a triangle into 4 congruent triangles, find the midpoints and construct segments from the midpoints to the midpoints. In squares, rectangles, and parallelograms, find the midpoints, and construct segments to opposite midpoints. An equilateral kite is a reptile, and an isosceles trapezoid has a neat subdivision. Regular polygons with 6, 5, 4, and 3 sides work well. It seems the less sides the better, and in most cases it helps to be regular.
From: LIMBERJ@mail.firn.edu Jaime Uhazie and Jenny Schaefer Grade 9 School: Martin County High School, Stuart, FL 1. Every triangle is different, so it depends on the triangle. 2. For a rectangle, draw a perpendicular bisector from the midpoints of each parallel side. This will split it into 4 congruent rectangles, and so on. For a square, do the same. For a parallelogram, do the same. 3. Another quaderlateral that would work well for a rep-tile is an isosceles trapezoid, because if you draw parallel lines, it splits it into similar trapezoids. 4. If a quadrilateral has a 90 degree angle, then it is a good candidate for a rep-tile because every angle will be congruent.
From: LIMBERJ@mail.firn.edu Julia Schumm and Edna Evans Grade 9 School: Martin County High School, Stuart, FL * The way to divide any triangle into four congruent similar triangles is to take the triangle and use a line to perpendicularly bisect one angle so it forms two 90 degree angles at its base. Then from the 90 degree angles put another perpendicular bisector forming another two 90 degree angles at that base. This is because you can infinitely bisect these 90 degree angles with perpendicular bisectors forming congruent triangles, because of the CPCTC theorem. * You can divide squares, rectangles, and parallelograms by cutting in half opposite angles to form triangles, then using the method above to infinitely divide the triangles into congruent triangles. * Other quadrilaterals are rhombuses, trapezoids. One kind of trapezoid which would have a neat subdivision is an isosceles trapezoid. * Other polygons are octagons, pentagons, and hexagons. These would be good candidates for rep-tiles, particularly if their angles are all congruent. * Any closed figure can be infinitely divided into congruent parts, using perpendicular bisectors.
From: Mungkee@aol.com Ben Ngo and Nick Tall Grade: 9 School: Martin County High School, Stuart, FL To divide any triangle into four similar triangles just connect each midpoint of each side of the triangle to each other. To make four similar polygons out of squares, rectangles, or other parallelograms do the same process as the triangle, connect the midpoint of each side together. Trapezoids work if you just connect the midpoints of the sides and not the bases. Other polygons do not work. To be a rep-tile you must have a triangle, a square, a rectangle, a parallelogram, or a type of trapezoid.
From: Brasscat5@aol.com Christine Francescani Grade 10 School: Martin County High School, Stuart, FL To divide a triangle into 4 congruent triangles you draw a triangle (any triangle), find the midpoint of each side, and connect the midpoints. With squares, rectangles, and parallelograms you also draw the shape, find the midpoints, and connect them. A rhombus is a special parallelogram and it can also be divided into congruent similar rhombuses by finding the midpoints and connecting them. I worked on the trapezoid division, but I couldn't find a way to make all of the shapes that were formed both similar and congruent. For other polygons I tried a pentagon, a hexagon, and an octagon. I could not see any way to divide them. As for a theory: When using a triangle, square, rectangle, or parallelogram, if you divide all of the sides at their midpoints and connect those midpoints, you will get a rep-tile. This doesn't encompass all polygons, but I tried my best.
From: lg123@ptialaska.net James Elgee Grade: 9 School: Juneau Douglas High School To split a triangle into four sections you make a triangle of three equal parts, all miniatures of the original with the center missing. You want the center to also be a miniature of the triangle only flipped vertically so that it fits in the gap made by the other three. That is how you divide a triangle into 4 congruent and similar triangles.
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7 December 1996