word lengths

a study by Rebecca Pitz, (part) published on the Grin website compares the word lengths in a variety of texts (extracts) and song lyrics:

results are as follows:

make some comparative comments

how does the range compare with the standard deviation?

3 x (median) = 2 x (mean) + mode is a relationship that supposedly works roughly - does it for this data?

Greek tragedies

of the many Greek tragedies that are thought to have existed around the 5th century BCE, it seems that complete texts are rare

only those from Aeschylus, Sophocles and Euripides are extant

the boxplot shows the word lengths from the written tragedies of these three writers

comment on the differences

eutrigon and co-eutrigon

in some novel and interesting work on the relationships between parts to wholes, Wayne A Roberts (Canberra, see the 'principles of nature' work) has established Pythagoras-like relationships for triangles that have one 60 degree angle

he has named these 'eutrigons' and triangles with one 120 degree angle (which he refers to as 'co-eutrigons')

he utilises unit triangles on an isometric grid
see the section on isometric areas on this blog

the relationships are straightforwardly derived from the cosine rule, substituting 60 degrees and 120 degrees but, importantly as far as appreciation is concerned, he presents justifications based on diagrams:

a 'eutrigon' is a triangle with one 60 degree angle

the eutrigon has equilateral triangles drawn on each of the sides

what is the relationship between their areas?

the triangles that look as if they are equilateral are equilateral

a dynamic version, created by Wayne Roberts:

for the co-eutrigon, this diagram is offered by Wayne Roberts:

prime number images

an interesting prime numbers graphic

shows the sum of the factors (listing all the factors in the sum) and the aliquot sum
tells you whether a number is abundant, deficient or perfect (or prime)
the chart can be dragged across to the left (see what happens at 360)

I have mentioned elsewhere my enthusiasm for using Alec McEachran's 'Primitives' application (that can be viewed full screen)

this screenshot shows the numbers 45, 46, 47 and (just) 48

composite numbers being shown in an interesting form











another representation shows the numbers on a grid, with the prime factorisation indicated by sectors

Steven Gerrard's goals

how Steven Gerrard's goals have been scored - up until the summer of 2012
what are the angles for the various sectors?

adapted (slightly) from the 'oh you beauty' blog
see also the 'fun with infographics'  section of the blog

frieze patterns

these Maori designs are good examples of frieze or strip patterns
powerpoint: frieze pattern types

the topic seems an interesting way of following up work on symmetry

Heather McLeay (Bangor University, North Wales)  has championed including a study of frieze (strip) patterns in the curriculum; her photographs of the seven different types are here

a frieze is an infinite strip of a repeating pattern, extending in one direction only
the symmetries talked about refer to the whole strip rather than sections of it

considering the isometries (distance preserving transformations): reflection, translation, rotation and glide reflection
and which combinations of these are not redundant
you can establish that there can only be 7 types of frieze pattern in the world

a glide reflection is where you reflect and then move the shape along a bit (like footsteps in the sand)

the Illuminations program allows generating shapes to be modified (by dragging) which might help understanding

there are several notations for classifying the 7 strip patterns and, once appreciated, students can be asked to analyse the symmetries and so recognise the types in various examples

the symmetries are of the whole (i.e. infinitely long) strip rather than just a small 'window' section

some symmetries take precedence in the notations

this notation is due to the IUC (international union of crystallography, there are (annoying) variations: sometimes 'a' is used instead of 'g'and sometimes the positions of the three symbols are different in different texts)
[the convention in these resources is from Wikipedia]

a clear youtube clip, geese

John Conway devised his own notation, based on images of the symmetries as various ways of 'stepping' (students could act these out)

an identification key

which of the 7 types are which?

for 2-dimensional design patterns:

the 17 possible wallpaper patterns
a wallpaper key

bar graph and chocolate

name that chocolate bar

from Arthur Buxton, the colours used in the wrappings and the proportions of their use are shown in a kind of percentage bar chart

which product goes with which graph?

Pie charts and Van Gogh

An artist, Arthur Buxton, has analysed the 5 most prevalent colours in 28 of Van Gogh's paintings (as well as those of other artists) and represented the extent of their use in pie charts:

which paintings are they?

starry night









self portrait

room at Arles

reaper

cubics

I think it's a shame that the cubic function doesn't get more of a look in in 11 to 16 education, in England anyway