By George Hart for the Museum of Mathematics

Paper is a versatile medium for experimentation and construction. Krystyna Burczyk has made beautiful sculptures by rolling paper squares to produce spiral corners that clasp together. They lock together without glue to create a kind of floral construction like modular origami but with spiral connections. This one has icosahedral symmetry—there are twelve purple locations with five-fold rotational symmetry. It isn’t traditional origami or kirigami, because of all the curves.

Below is a second example, this time with octahedral symmetry. There are six four-fold rotation points and eight three-fold rotation points.

An additional example below displays simple eight-fold rotational symmetry, making clear how the basic joint works.

There are many more examples in her galleries here and here. Krystina calls them simply “twirls,” but I like to call them “twirligami.”More:

## 5 thoughts on “Math Monday: Twirligami”

1. Thai Nguyen says:

While both had amazing international ti le ca cuoc chau au and club careers and are right up there with the best of the last 50 years it could be said they never fully fulfilled their potential given their luck tỷ lệ cá cược bóng đá trực tiếp with injuries, their fondness for the odd nightclub and chomping down on the odd burger or seven….

2. nhan says:

Bet on Both Teams to Score Strategy

The first thing you cuoc bong da tren mang want to look for in markets like this is teams who are attack minded. Remember you need a team that is good going forward, but also capable to concede the odd goal. This is why it’s not always going to be best to bet on the bigger teams such as possibly a Manchester United kèo tỷ số bóng đá or Barcelona as, whilst they are more than likely going to score against the majority of opposition, they are teams that aren’t going to concede many goals either. In fact, on the flip side you should be looking at teams keo ty so truc tuyen that are lower in the league and even near the bottom who are likely going to struggle to keep clean sheets but are capable of nicking the odd goal.

3. nhan says:

After a spell in the Old Trafford kèo da banh cold before a Deadline Day move to Real Madrid collapsed, reports suggest that Louis van Gaal might finally be ready to bring David de Gea back into his warm embrace.

Writing in the Daily Mirror, ti le da bong David McDonnell says that Van Gaal is set to recall De Gea for the visit of Liverpool this weekend. The Dutchman left out his No. 1 goalkeeper in favour of Sergio Romero tỷ lệ cá cược tỷ số whilst the sorry business of his attempted move away was sorted out (or not).

Van Gaal will reportedly hold clear-the-air talks with De Gea when he returns from the international break. Praise the shambolic administration!

4. nhan says:

Looking at the profit/loss (P/L) summarie độ bóng đá online in the ‘Totals’ column, adding together the first six rows of odds clusters produces a loss of -2,564 units, based on a flat stake of 100 units per bet.

Essentially this means if the away team was priced hướng dẫn cá cược bóng đá as a clear favourite or close to the home team’s prices, they won less frequently than the probabilities indicated by their odds. The last of these first six cluster groups closes at away odds of 2.90.

Look at the second row of the table. The odds cluster between 1.66 (implied probability 60.2%) and 2.00 (implied probability 50%) contains 83 matches and, if the odds had been ‘fair’, 55.1% (60.2% + 50% / 2) of the away teams priced in this group should have won.

As you can see, this was not the case! Of 83 games in five seasons only 43 were away wins (51.8%).

Therefore, punters huong dan ca cuoc bong da tren mang who regularly backed away favourites in the Bundesliga during 2010-15 surrendered ‘value’ in their bets to the bookmakers. When this happens, only one side of the deal wins in the long-run; invariably it isn’t the bettors!