The Riach Wallpaper Frieze

 

The Background

During the checking of homework with 5X2 difficulties had been found with differentiating:

y = sin2u + cos2u

Since the example occurred in an exercise concerning the use of the 'chain rule', naturally that was what was used. So the thinking went:

y = (sin u)2 + (cos u)2

and when differentiated this gave:

2(sin u)(cos u) + 2(cos u)(-sin u) = 0

a somewhat surprising answer.

The Discovery

What sort of graph has gradient 0? The class looked sullen and uninvolved.

"A flat graph" answered Chris, beginning to get involved in the proceedings.

But how can a graph with a formula involving sines and cosines be "flat"?

Chris, now feeling challenged, was rising from his usual slouch to a more upright position. No comment was forthcoming.

"sin squared plus cos squared?" said the teacher in hope of some vague recognition.

Chris was now bolt upright in his desk and the signs of deep thinking were beginning to show on his face. Suddenly a flash of insight surged through his awakened brain. " It's to do with the graphs... the heights...one's going up and the other's going down...and they sort of...." Words failed him as the graphical vision overwhelmed him.

The teacher, struggling to understand another's vision, suggested a closer look at the graphs:

To obtain the graph:

y = sin2x

square the y-coordinates of points on the graph:

y = sin x

To obtain the graph:

y = cos2x

square the y-coordinates of points on the graph:

y = cos x

"Ah Ha!" said the teacher "You mean they sort of balance out?"

"Yes. that's it!!" cried Chris and an expression of satisfaction settled on his face drying the beads of perspiration on his brow. "They'd fit together".

This last comment was a stroke of genius. "You mean like a wallpaper frieze?" but the teacher's words were muffled by the clouds of coloured chalk dust that were now billowing from the frenzy of activity on the blackboard. The teacher finally emerged from the cloud and as the dust subsided a work of art was revealed:

y = sin2x
y = cos2x
y = sin2x + cos2x

"What is sin2x + cos2x always equal to?" The teacher glared menacingly at the class.

"One" they all replied in unison and fear.

"And what does the graph y = 1 look like?" The teacher was pointing at the revealed work of art on the board. Around the classroom slow waves of understanding ebbed and flowed over the sea of faces. "And what is it's gradient?"

"Zero" resounded from wall to wall.

"And whose going to make a fortune with his wallpaper design?"

"Chris Riach!" The class gazed in admiration at this new budding entrepreneur that had emerged from their midst.

The Riach Wallpaper Designs