REGULAR TILINGS

Description A regular tiling is a tiling of the plane consisting of multiple copies of a single

regular p olygon meeting edge to edge Howmany can you construct

Comments While these tilings can b e easily found by trial and error and all are quite

familiar they do give some insightinto the geometry of regular p olygons

Hints There are only three regular tilings

References Kappra Connections The Geometric Bridge Between Art and Science

McGrawHill Chapter

REGULAR TILINGS

Illustration Kappra p

SEMIREGULAR TILINGS

Description A semiregular tiling is a tiling of the plane consisting of at least two dierent

typ es of regular p olygons In addition the pattern of p olygons meeting at a common juncture

p oint or vertex must b e the same for each p oint How many can you construct

Comments Some of these tilings are not immediately obvious You can calculate or

eliminate some p ossibilities by making a table of the measures of the interior angles of

regular p olygons since the sum of these angles for the p olygons meeting at a common vertex

must b e degrees

Hints There are only eight semiregular tilings

References Kappra Connections The Geometric Bridge Between Art and Science

McGrawHill Chapter

SEMIREGULAR TILINGS

Illustration Kappra p

PLATONIC SOLIDS

Description A Platonic solid has the prop erty that eachfaceisanidentical regular p oly

gon and that the same numb er of p olygons meet at each corner Can you construct them

all

Comments These are ancient ob jects of study Plato naturally is reputed to have

investigated them and they are discussed in Euclids Elements but they may b e quite older

Hints There are only ve Platonic solids

Tetrahedron Four equilateral triangles three meeting at each corner

Cub e Six squares three meeting at each corner

Octahedron Eight equilateral triangles four meeting at each corner

Do decahedron Twelve regular p entagons three meeting at each corner

Icosahedron Twenty equilateral triangles ve meeting at each corner

References

Holden Shap es Space and SymmetryDover

Wells The Penguin Dictionary of Curious and Interesting GeometryPenguin pp

PLATONIC SOLIDS

Illustration Wells p

SEMIREGULAR SOLIDS

Description A semiregular solid has the prop ertythateach face is a regular p olygon but

that there are at least two dierent kinds of faces In addition the same pattern of faces

must meet at each corner How manycanyou construct

Comments According to Heron and Pappus Archimedes wrote ab out these solids but

unfortunately the work has b een lost These are b eautiful ob jects with very app ealing

symmetry

Hints Taketwo copies of any regular p olygon and use them as the top and b ottom of a

drum ringed with squares This is a prism Now remove the squares and use a strip of

equilateral triangles to form the sides of the drum This is an antiprism There is an innite

numb er of prisms and antiprismsone for eachchoice of base p olygon Apart from these

there are exactly thirteen semiregular solids also called Archimedean solids One of them

is made with hexagons and p entagons and is the template for a so ccer ball as well as a

representation of a recently discovered form of carb on C known as Buckminsterfullerene



References

Holden Shap es Space and SymmetryDover

Pearce and Pearce Polyhedra Primer Dale Seymour pp

Wells The Penguin Dictionary of Curious and Interesting GeometryPenguin pp

SEMIREGULAR SOLIDS

Illustration Wells pp PearcePearce p

THE DELTAHEDRA

Description A deltahedron is a p olyhedron made solely from equilateral triangles There

is no requirement that the same numb er of triangles meet at every corner but the ob ject

must b e convex ie it should have no indentations and two triangles that meet along an

edge must not lie in the same plane Howmany can you construct

Comments It is conjectured that the name comes from the fact that the Greek letter

lo oks like a triangle

Hints Try to rst understand whyevery deltahedron has an even numb er of faces Exp er

iment to see if you can determine the minimum and maximum numb er of faces Curiously

one p otential deltahedron is missing

References

Holden Shap es Space and SymmetryDover p

Pearce and Pearce Polyhedra Primer Dale Seymour p

THE DELTAHEDRA

Illustration PearcePearce p