### 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?

### Submissions

```From: visitor1@sasd.k12.pa.us
Amy Saveikis, Renee Tappe, Maria Briski
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

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)

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

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
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
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
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
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
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
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
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.

James Elgee
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.

```