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All about **ARCHIMEDEAN SPIRALS** and **GOLDEN SPIRALS** – with tips on how to draw each either by hand or on a computer.

**FIVE FINGERS:** on each hand, with two hands: this is why we count in tens! – starting a new column each time we get to ten in the previous column. One of the simplest ways of keeping count is to use a **tally chart**: e.g. if you were counting the number of cars of each colour, you would draw one line for each car, then use the fifth line to “close the gate”.

**A JOKE FOR YOU!:** an ancient Roman walks into a bar, holds up two fingers, and asks “five lemonades, please”.

Do you get the joke? HINT: the ancient Roman symbol for five is the letter V.

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NOTE: this blog is limited to flags of Sovereign States (“nations” – and two “territories” that were just too cool to miss out), and is not concerned with any politics / symbolism in flags, it’s purely about their shapes and colours!

**5) NEITHER LINE NOR ROTATIONAL SYMMETRY e.g. NEPAL: **no symmetry at all, and the only pentagonal flag (5 sides – count them!) – it’s not even rectangular! It also features a 12-pointed star and a partly-hidden 16-pointed star. So cool! Also pictured are the flags of Bhutan (white dragon), Seychelles, Wales (red dragon) and Kuwait (green, black and red trapezia + a white rectangle), none of which have line nor rotational symmetry.

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ALL OF THE SHAPES IN THIS IMAGE ARE RHOMBI (another word for “rhombusses”) BUT ONLY THOSE LABELLED 2 ARE DIAMONDS.

In the image: shape number 1 (purple) is a **Square **(not a diamond), because it is a **quadrilateral** (4 sides) with four $90^{\circ}$ **right-angles**. The fact that it is “turned around and balanced on a corner” does not change the fact that it’s a square. Non-mathematicians will often say this is a “diamond” because it’s balanced on a corner. But it’s not a diamond: it’s a square!

**DIAMOND:** The shapes labelled number 2 (blue, red, green) are all **Diamonds**, because they are each made up of two equilateral (all sides the same length) triangles stuck together. I’ve emphasised this by drawing a dotted line between the two equilateral triangles in the blue diamond.

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**A WARM-UP:** Quickly: is 91 a prime number or not?

Don’t think too hard, just your gut feeling: yes or no? I’ll return to this later, but first:

**NINETY-ONE IS A TRIANGULAR NUMBER:**

The **Triangular numbers** **1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, …** come up in lots of places, but at their simplest they are the number of snooker balls you would need for a triangle with, in this case 13 rows. Ninety-one is triangular because 91=1+2+3+4+5+6+7+8+9+10+11+12+13. So we could use the notation $91=T_{13}$ because it’s the thirteenth Triangular number.

**NINETY-ONE IS A SQUARE-BASED-PYRAMID NUMBER:**

The SQUARE PYRAMIDAL NUMBERS **1, 5, 14, 30, 55, 91, …** are the number of cannonballs you would need in a stack of nested squares with, in this case 6 rows. Ninety-one is Square-Based-Pyramidal because $91=1+4+9+16+25+36=1^2+2^2+3^2+4^2+5^2+6^2$.

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HOLOGRAPHIC TAPE in choice of Blue, Silver, Green, Red, Gold, Magenta (please specify when ordering).

STRETCHY BAND is always red / orange.

Every order comes with an INFORMATIVE INSTRUCTION SHEET complete with diagrams.

Please see our current price list. UK orders only at time of writing.

Please contact House of Maths or email us directly (email address on price list) with quantity required, chosen colour option(s) and postal address. We will then email you back with our Bank Details. Once we have received your payment, ShΔpe-Shifter is made to order and dispatched within 3 working days.

**THREE** is the corners (or sides) of a triangle, the somersaults in a triple and the wheels on a tricycle. Three is the notes in a triad, the musicians in a trio, and the tricks in a hattrick. Three is the Musketeers in Dumas’ classic novel. Three is the only number which is both **prime **(only two factors that go into it) and **triangular **(1+2). **Trigonometry** – the huge branch of mathematics dealing with sin, cos and tan, literally means “measuring triangles”.

**TWO BABIES PUZZLE: **two babies are born to the same mother on the same day of the same year at the same hospital, yet they are not twins. The explanation?:

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A big thank you to St Martin’s School for this fabulous memento of Shapes Day 2023.

Here’s what a House Of Maths visit might look like at a small (one form entry) Primary school. The day featured a Whole-School Shapes Assembly followed by seven different 30-minute workshops – one with each class from Reception to Year 6. The workshops were all shapes-themed, with every child doing the maths themselves. Featuring tesselations, special quadrilaterals, 3D shapes with vertex notation, AB and ABC patterns and much more.

More info and bookings at houseofmaths.co.uk

I was super excited in Summer 2022 to visit Maths City: the UK’s first Interactive Maths Centre! Located in the Trinity Shopping Centre Leeds, Maths City is crammed full of fun, colourful objects to pick up and play with. Notably absent are… numbers! Almost all of the exhibits use shapes, mirrors, bubbles and even lasers: but rarely numbers. This was a conscious decision, explain Maths Funsters Dan, Vittoria and Jerry,

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How do we put into words what we mean by a “square”? Here are six naughty shapes pretending to be squares. But they are all fakes! Can you see what’s wrong with each **bad definition**?

1) SQUARE: A SHAPE WITH FOUR SIDES OF EQUAL LENGTH: but hang on – this shape is not flat! It just goes around four edges of a cube. Hmm, let’s try another definition:

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