Personal Blogs

"I love camping", said Tom intensely.

What’s the most difficult tongue twister? It’s hard to say.

My brother photographed this astonishing lichen, Cladonia pyxidata, or Pixie Cups.

I used to think nostalgia meant a kind of sentimental hankering for times gone by, our own times or a past age of our romantic imagining. But it seems that when it first came into English in the 1700s it meant homesickness, derived from Ancient Greek νόστος nostos = going home and αλγία algia = pain (as in neuralgia for example), and it only came to mean longing for a time rather than for a place during the last century. 

The modern Greek word νόσταλγία does indeed refer to homesickness but I haven't been able to find with certainty whether this has come directly from Ancient Greek or is a back-borrowing from English or perhaps French,

At any rate it is a beautiful word, with a lovely wistful feel to it, and its original meaning of homesickness is the one I prefer.

“Just turn where I say, and you won’t get lost”, said Tom forthrightly.

Gaspard Monge was a French mathematician of the 18th and 19th century, a man of many parts who was a friend of Napoleon and took part in the latter's expedition to Egypt. He gave his name to a theorem in geometry which is easy to see in this illustration from Wikipedia (image by Jason Quinn).

Although it is usually presented as as theorem about circles and tangents it seems more general than that. Here is my drawing of an analogue for equilateral triangles, produced using GeoGebra.

For any pair of triangles, the lines drawn through pairs of corresponding points meet at a point, and for any set of three triangles taking the triangles in pairs produces three such points that all lie on the same line.

I don't think equilateral triangles are special, as far as I can see we could choose any shape.

Monge's theorem can also be generalised to three dimensions (four spheres and six cones, one for each pair of spheres, all of whose vertices lie on a common plane), and indeed to higher dimensions, although it obviously gets harder to visualise.

Herb and Spice-themed Daffynitions

Anise: Sibling's daughter.

Borage: Jail term.

Chervil: Type of rodent, often kept as a pet.

Chicory: Deception.

Cloves: Fings you wear.

Cumin: Opposite of going.

Fennel: Tapering tube for guiding liquids.

Mace: Small rodents.

Marjoram: Impaired your molasses-based alcoholic spirit.

Rosemary: Got up in a cheerful mood.

Sage: Reason why I'm not as sprightly as I once was.

Thyme: See Borage. 

Our old friend Tom has been at it again.

After I'd thought of that one I wondered if Chat GPT knew about Tom Swifties. It does and came up with quite a decent effort when prompted with "pie" and "Tom Swifty".

Some more geometry! 

We can inscribe a square in an equilateral triangle so that all itas corners lie on the sides of the triangle. How? Well, consider this sketch.

We draw a chord of the triangle parallel to the base and draw a square with the chord as one of its sides. Then we move the chord vertically downwards, keeping it parallel to the base and with its ends on the sides of the triangle and progressively enlarging the square, until the base of the square lands on the base of the triangle, and we are done.

We can extend the idea to three dimensions and inscribe a cube in a regular tetrahedron.

We take a section parallel to the base and in this equilateral triangle inscribe a square using the method described above, and construct a cube with this square as one of its faces. Then we move the section vertically downwards, keeping its vertices on the sides of the tetrahedron, progressively enlarging the cube, until the base of the cube lands on the base of the tetrahedron, and we are done.

It's conjectured, but I don't think proved, that this gives the largest cube that can be inscribed in a regular tetrahedron, see here.

Rather neat, and It seems to me that an analogous construction would be possible in dimension and beyond, a hypercube in a 4-simplex and so on, but I can’t prove it and drawing a picture of even the four dimensional case feels quite a challenge.

This attractive little plant likes my front garden wall, where it grows profusely. In the photo it’s intertwined with some actual ivy, at left, and you can see that the leaves of the toadflax really do look like miniature ivy leaves.

This year there is more of the toadflax than I can remember ever seeing before. Perhaps it’s because we have such a lot of rain in the last few months.

🍇

If you try to steal my grapes, then you better watch Shiraz.