KNOW THYSELF

 

KNOW THYSELF

 

The following is a meditation that anyone can use because every person has these qualities.  You have these qualities.  Whether you think of these as “fruits of the Holy Spirit,” innate Buddha qualities, or something else, it is worthwhile to get acquainted with yourself, and ponder your purpose.  Read the following list as a meditation about yourself.  Ask yourself what you would add to this list.

 

1. I have a song so I can be singing.

 

2. I have serenity so I can be peace-filled.

 

3. I have detachment so I can be centered and calm.

 

4. I have lightheartedness so I can be relaxed.

 

5. I have inner joy so I can be joyous.

 

6. I have a sense of humor so I can be laughing.

 

7. I have playfulness so I can be amused.

 

8. I have mindfulness so I can be aware.

 

9. I have perception so I can be discerning.

 

10. I have wisdom so I can be wise.

 

11. I have intelligence so I can be enlightened.

 

12. I have insight so I can be intuitive and seeing beyond appearances.

 

13. I have creativity so I can be creating.

 

14. I have sensitivity so I can be considerate.

 

15. I have empathy so I can be empathetic.

 

16. I have thoughtfulness so I can be kind.

 

17. I have compassion so I can be compassionate.

 

18. I have gentleness so I can be caring.

 

19. I have generosity so I can be giving.

 

20. I have gratefulness so I can be appreciative.

 

21. I have patience so I can be patient.

 

22. I have courage so I can be continuing.

 

23. I have steadfastness so I can be steadfast.

 

24. I have fortitude so I can be strong.

 

25. I have perseverance so I can be overcoming.

 

26. I have hope so I can be confident.

 

27. I have willfulness so I can be determined.

 

28. I have watchfulness so I can be vigilant.

 

29. I have freewill so I can be choosing.

 

30. I have faithfulness so I can be loyal.

 

31. I have integrity so I can be honest.

 

32. I have a sense of fairness so I can be just.

 

33. I have awe so I can be reverent.

 

34. I have a deep well so I can be loving.

 

35. I have goodness so I can be amazed at me.

 

36. I have wholeness so I can be complete (holy, whole, sacred).

 

37. I have inquisitiveness so I can be seeking.

 

38. I have a journey so I can be discovering.

 

39. I have a sense of direction so I can be on track.

 

40. I have trust so I can be walking on water.

 

I have opportunities and abundance so I can be expressing me.  I have many gifts so I can be becoming. 

 

-2008-

 

Slide show, music, and folders on my main page.

 

 

---

 

 

 

Full-size image

 

 

Spirit of Wisdom,

Peace, and Renewal,

enlighten us

and bring harmony

to our lives,

and make the whole world

holy.

 

 

 

SIGNS OF THE TIMES

 

 

SIGNS OF THE TIMES

 

“Is there anyone at your house over 65 who could get a senior discount?” she asked slyly looking at my gray hair.

 

Over 65!  I thought the senior discount was for anyone 55 and older.  That was at the medical supplies store where I was buying a hot water bottle to keep my toes warm this winter, when I plan to have the heat turned down to save energy.  Is this store pinching pennies?  Well, most of their customers would be seniors and they can’t be expected to give a discount to all of them.  (By the way, if you use a hot water bottle, you learn very soon to wrap it securely in a plastic bag in case it leaks, and to not let the water be too hot.)

 

Then at the grocery store I noticed that the produce department had shrunk.  Well, harvest time is over isn’t it?  Taking the place of produce was a huge display of carbonated beverages.  But I noticed that the baked goods section had also shrunk.  (This grocery store has its own bakery – very nice.)  Actually, I’d noticed that before – the baked goods area has been steadily shrinking for several months as they remove sales tables and counters.  The case of the shrinking grocery store (hmmm).

 

No problem finding everything I wanted in the grocery store this time, unlike previous visits over the past few months when some things were out of stock.  I kept asking myself how this store could be having a problem staying stocked just a mile or two from one of the biggest interstates in the nation, if not the biggest.  Producers are faltering?  And now there is more to buy, because others are buying less?  I don’t know what’s going on.  This store carries some organic products, so I go there for the organic.  I need this store.

 

And what’s happening with the salad bar?  Seems like the new policy is to make sure each and every bin is nearly empty before they toss what’s left and replenish.  I got a well-picked over salad.  Oh well, maybe the salad bar was just unattended for some reason?

 

There was something else new – two huge tables filled with assorted items, all sale-priced 99 cents or less; all discontinued brands I suppose.  Not great on a day when the papers were screaming “global meltdown” or some such.

 

Another thing I like about my grocery store is that they take the groceries to the car and load them.  What do you say to a man pushing your groceries?  I asked this man “What do think about the ‘crisis.’  Is it real or are they just trying to sell newspapers?”  He had plenty to say on that – he is retired he says, in his seventies, like his wife.  (Some retirement where he has to keep working!)  He said, “I don’t trust the banks; I don’t trust Washington.”  He’s worried about his very small retirement savings.  He didn’t say he’s angry and scared, but he is. 

 

I am worried that in these hard times this grocery store could be losing customers.  It’s such a great store, I hope it can stay open.

 

Earlier that day I noticed the gas station had changed, too, and I don’t mean the sky-high prices for gasoline.  For some months now, the new rule has been cash customers pay inside first unless they are at the nearest pump where they can be observed by the checker.  Dishonesty, or rather desperation, has been growing along with the prices.  Now I noticed a new sign – “For these credit cards, pay inside” (they named two card companies).  So they don’t trust those cards anymore. 

 

Another sign of the times – almost no traffic!  I was thinking maybe I was the last person on Earth.  People are finally learning how to conserve gasoline and to do more than one errand at a time?

 

Many houses for sale on my street.  Where will the people go? 

 

Later that week the local paper announced some teachers might be fired – not as many children had signed up as anticipated.  Where are the children?  Another article mentioned 500 homes on the market locally that were foreclosed (banks took over).  Could the lack of children have something to do with that?  People are moving to less expensive areas?   Nobody knows what’s happening with the schools.

 

Several weeks ago the clerk at the drug store said that more and more people are asking for rain checks (to get the current price at a later date).  Meaning more things are out of stock?  Or just more people are asking?  Or both?  I don’t know.

 

When I went to the bank in early August, the desk clerk there said, “Watch out for that other bank,” and he named a bank.  “It will fail.”  He said he was sure.  He said it’s easy to find out how sound a bank is by going online.  At the time I thought he was just trying to steer me away from his competition and shrugged it off.  Now if he knew two months ago what was going to happen to that particular bank, and obviously the bank’s officials must have had a pretty clear idea of what could happen, why didn’t the government know AND do something well in advance of this crisis to head it off?

 

Why hasn’t the government done something about health care, immigration, global warming, etc?  This Congress was elected to stop the war.  They didn’t.  Did they do anything?  Oh well, they kept the government running.  That’s good.

 

The bank clerk said, “If you want to be secure, buy a cow.”  Now I’m thinking maybe this guy knew what he was talking about.  Now I’m thinking a cow would be a good idea, except that a cow would be too big.  Maybe I need a few sheep instead.  They would take care of cutting the grass and that way I wouldn’t have to pay for lawn care.  Sheep milk tastes OK, doesn’t it? 

 

My view – when people’s lifetime savings can vaporize overnight or in a week, just because other people are gambling, and that’s what it is, gambling in the stock market with nearly real-time online trading, and gambling by making bad bank loans, then this gambling has got to stop.  We have an economic system that is badly in need of radical reform.  Forget about just rearranging the deck chairs.

 

-2008-

 

Slide show, music, and folders on my main page.

 

X Keywords:   Financial MSN Windows Live Spaces X

 

THE BIRD FEEDER TEST

 

THE BIRD FEEDER TEST

 

In fairy tales, the young man trying to win the hand of the king’s daughter must perform some difficult feat, like killing a fire-breathing dragon, before he can marry the princess and become the heir to the throne. 

 

Shouldn’t we ask our political candidates to show us what they can do; shouldn’t they have to pass some test before they are allowed to run for office?  After all, being able to spend campaign money handed over by corporate donors hardly demonstrates one’s proficiency at governing, nor does being part of the government a specified number of years.  Well, of course I know that no test would give us the answers we need and besides, testing wouldn’t be democratic, but what if >>>

 

What if each candidate was required to perform some simple task, like putting up a bird feeder, keeping it filled with seed, and doing it all so the squirrels did not get the seed?  We’d watch them at their tasks like we watch the characters on reality TV.  We’d watch them go the big box store, and see what type of bird feeder equipment they select, or we’d watch as they made a homemade bird feeder.  We’d learn whether they are dedicated enough and patient enough to feed the birds every day, whether they are smart enough to be able to outwit the squirrels.  We’d give them only so much money and see if they could live within a budget.

 

We’d take off points if a candidate buys a plastic bird feeder and the squirrels gnaw big holes in the feeder.  We’d take off points if the squirrels are able to climb onto the feeder and empty it more than once a day.  Extra points off if he allows the seeds that tumble down onto the ground to stay there and attract mice.  Extra points off if the mice population starts to attract snakes.  He’s out of the running if any snakes or mice come into the house.  If the raccoons knock down the bird feeder and start batting it around the yard to get out the seed, it’s an automatic F.  He’s disqualified if deer come and nibble at the seed and leave deer ticks all over the yard.

 

If the candidate starts shooting at the squirrels with a gun to kill them, he’s not the right kind of candidate for us.  We need someone who can manage to accomplish things without resorting to violence. 

 

What if the candidate decides he needs more bird seed and decides to “liberate” some seed from the neighbor’s bird feeder?  What if he goes over to the neighbor’s yard and starts shooting the neighbor’s squirrels, shooting the birds, mice, raccoons, and deer and anything else that is trying to get at the neighbor’s bird seed?  What if he wrecks the neighbor’s yard?  What if he then insists he will not leave the neighbor’s yard until he has gathered up every last bit of bird seed over there?  Maybe some imperialist tendencies?

 

Now I’ve been saying, “he” throughout my musings here, but of course, a candidate could be “she” just as well. 

 

I think I’d be impressed by a candidate who could figure out a way to thwart the squirrels, maybe by putting the bird feeder up on a squirrel-proof metal pole.  He’d get points for having it close enough to trees so the birds could fly there to rest and feel secure, and yet far enough from trees so the squirrels couldn’t jump from the trees to the feeder.  High enough so passing deer couldn’t get seed and passing cats couldn’t get at the birds.  With a base that could be swept with a broom. 

 

He’d get even more points by doing it all in an ecologically sound way (with organic bird seed, for example), with calm, thoughtful planning.  I’d be even more impressed if he also found a way for all the wild creatures to share the yard and live together in peaceful harmony.

 

Life hands us plenty of unanticipated happenings, consequences, and surprises each time we start a project, each time we try to accomplish anything.  Somehow we need to know not just if a candidate can accomplish a particular task such as installing a bird feeder – that part is relatively easy; we also need to know how he will handle all the unanticipated happenings (like mice and snakes materializing under the bird feeder), and if he is smart, analytical, flexible, carefully deliberative, and capable of crafting a strategy that is responsive, fits the moment, and will work in the long-term.

 

It would be fun to watch the squirrelly misadventures of the candidates on reality TV.  Better there than on the nightly news after they get elected.

 

Look before you leap.

Full-size image

 

I’d like to get the current candidates into some sort of Survivor setting where we’d quickly learn if they could in fact work cooperatively together to accomplish some positive result.  What little I saw of Survivor was a disappointment (before I stopped watching programmed TV many years ago).  None of the show’s contestants moved beyond the boundaries inherent in the show and they did little but form “alliances” and generate sound bites.  We need to know if our candidates could do better than that, taking initiative, moving beyond boundaries, demonstrating an ability to overcome gridlock in DC. 

 

In Survivor, the outcome is never in question – one person is left.  A real surprise ending would be if the contestants banded together, refused to compete, and persuaded the show’s executives to distribute the prize money equally among all the contestants.  (Do we ever step outside the script for our own lives?)

 

-2008-

 

Slide show, music, and folders on my main page.

 

X Keywords:   candidates for election MSN Windows Live Spaces X

 

IT’S ALL HAPPENING IN MY GARDEN

 

IT’S ALL HAPPENING IN MY GARDEN

 

My garden is starting to look like a September garden.

 

Many of the marigolds have started going to seed.  I’ve started collecting the seed.  I’ll likely get enough to fill a bucket.

 

Did you know you’re not supposed to plant small marigold plants (several inches high with small crimson and orangey-yellow flowers) in the same yard with large marigold plants (much taller with large lemon-yellow flowers).  You’ll get a hybrid that is a large woody plant with small crimson and orangey-yellow flowers.  Well, I did that.  I didn’t suspect anything would happen.  Now I have almost none of the regular large marigold plants left.  What I have are maybe fifty or a hundred of the hybrids, some of which are very tall.  Mostly more than knee high.  A few chest high!!!  The tallest are growing where I have compost.  The hybrids tend to be very bushy in shape.

 

I do think that the hybrid is an improvement over either of its predecessors.  There are far more flowers per plant.  And while I guess I prefer lemon-yellow, I don’t really care which.  After years of planting this hybrid I guess I have a plant that is very well adapted to the hard clay in my yard (heh, heh).

 

Let me add here that I have the marigolds planted partway along two sides of the house so you don’t start visualizing whole fields of them.

 

I’m glad things turned out the way they did.  I guess the moral of the story is this – things could turn out better than you could ever imagine.

 

Or not?  Well, here I am disturbing the balance of nature big time in my garden.  The tall hybrids are ideal habitat for praying mantises who like:  (1) to be a few feet up off the ground, and (2) to be able to snap at any flying insects coming to the flowers.  Which means that my lovely garden is not so butterfly-friendly. 

 

I’ve been conserving mantis egg cases when I find them.  Of course they are beneficial insects, keeping the garden free of pests.  But I think that this year I will be moving all the egg cases out of the marigold area into some other area of the yard that has no flowers blooming later in the season when the mantises are big enough to prey on the butterflies.

 

Whether that will work or not I don’t know.  Mantises can fly.  For sure many will still find their way to the marigolds.  Which is good because I need a few?  But not too many.

 

I wish the praying mantises could kill the Japanese beetles which are voracious pests.  I found out why they can’t.  I watched a standoff between a Japanese beetle and a praying mantis many weeks ago.  The mantis moved very, very slowly in the direction of the beetle that just sat facing the mantis.  This inching forward took perhaps a minute or two.  Time stood still.  Then faster than the eye can see, the mantis struck.  Thud!  Then – nothing.  The beetle’s hard shell was impenetrable?  The mantis wandered off.  The beetle, realizing the coast was clear, stretched out its many legs (that had been curled under it for protection?), and started to wander off, too.  Interesting that the beetle could sense the mantis, curl up, and protect itself.  Interesting that the beetle could not defend itself against me.

 

I should add that the hybrid marigold is fairly resistant to the Japanese beetles, unlike its large marigold predecessor which would get demolished.  The beetles like to sit on top of the hybrids, but don’t seem to damage them.  And when they’re all congregated like that, makes it easy to spot ’em.

 

I’ve been trying to change the balance of nature by weeding out the trumpet vine (with small trumpet-shaped vermillion flowers) that grows everywhere and makes a tangled mess.  But this year I planted morning glories, and somehow ended up with more trumpet vine in that area than morning glories.  I think because the leaves are so similar, I left the trumpet vine thinking it was morning glory. 

 

But that turned out alright anyway.  The hummingbird likes the trumpet vine flowers – a lot.  I can look out my window over that bit of garden.  Once in a while the hummingbird will come zooming in, then dart at the flowers for nectar.  One time I was watching, I saw it perch a couple of times on plants to rest – I never saw that before.  (The hummingbird does not perch while feeding.)  I wondered if the hummingbird knew pure happiness, perched there in quiet sunshine surrounded by flowers.  What is even better is to be out in the garden and have a hummingbird come by close enough so I can hear its wings humming.

 

I very much like to see the goldfinch perched on the tall coneflowers, busily pecking out the seeds.  The goldfinch is not really gold, but lemon-yellow with black trim.  That’s the male.  The female has a greenish cast to her back and so is closer to gold.  I had almost weeded the coneflower thinking it was a weed.  Well, it is a weed but I’d intentionally put the coneflower seed there, seed I’d gathered from my drain field meadow.  Just didn’t recognize it at first.  That turned out well.

 

When I saw my second katydid (strange insect that looks like a leaf), I wondered at the marvelous balance of nature that allows the katydid chorus to play in the evenings.  The katydids are mostly unseen up in the trees, playing their tune.  It is such a soothing cadence.  And they do it without any help from me.

 

-2008-

 

Slide show, music, and folders on my main page.

 

X Keywords:   Garden MSN Windows Live Spaces X

 

ONCE UPON A TIME

 

ONCE UPON A TIME

 

“It is the Earth Goddess for all to see” – that’s what Silbury Hill is according to the narrator on the DVD, “Goddess Remembered.”  It is “the tallest prehistoric man-made mound in Europe and one of the world’s largest,” according to the Stone Pages site on antiquities.  It looks like a gigantic pile of dirt.  It is more than four thousand years old.

 

I couldn’t recall Silbury Hill.  I looked it up in my book, Voyages of the Pyramid Builders; The True Origins of the Pyramids from Lost Egypt to Ancient America, by Robert M. Schoch, Ph.D. with Robert Aquinas McNally.  The book says, “Whiteness is a key characteristic of another pyramid candidate:  Silbury Hill, which is located just south of the village of Avebury in England’s Wiltshire.  A massive artificial mound with a flat top, Silbury Hill stands approximately 130 feet high, with a base circumference of 1,640 feet that covers more than five acres.  The hill contains over 12 million cubic feet of earth and chalk, the latter originally coloring it white.  This monument was built in three stages, the first beginning in about 2660 BC.  Various legends and stories attach to Silbury Hill, but the monument’s purpose remains unknown.”

 

If Silbury Hill is a mound of the Great Goddess, does it signify her breast or her womb?  The DVD would seem to favor the idea of a breast because it launches into some sort of paean to her breast.  However, a BBC article says “Earth worshippers believe that Silbury is, in fact, the swollen womb of the Earth Goddess and provides a symbol linking the harvest with the pregnant earth.” 

 

Silbury Hill is flat on top, but that flat area is atop some sort of round crown, a circle within the much larger circle of the base circumference.  And the round crown – is it a nipple or just some erosion?  This tourism page says, “Each of the six steps was concealed within the overall profile of the mound, except the last one at the top which was left as a terrace or ledge about 17 feet (5 m.) below the summit. This terrace is clearly visible on the eastern side of the mound, but less distinct from the west.”  Yes, it is clearly visible and apparently intended – so the breast idea seems plausible.  I’ll add here what the UK’s English Heritage says about Silbury Hill, “Its purpose and significance remain enigmatic,” just so you know what the official position is.

 

According to the DVD, people dug into Silbury Hill in order to search for ancient treasure, but apparently, there was no treasure (at least no known report of it).  The top of Silbury Hill collapsed in 2000 and this was a delayed result of destructive excavations in 1776.  And other excavations at Silbury Hill have caused similar problems.  (Who would dig destructive tunnels into an ancient artifact!!!  What a shame!!!)  And now as part of an effort to analyze and remedy the problem tunnels, “investigators” have drilled seven boreholes deep into the mound.  No doubt adding to the destruction.

 

The DVD shows Silbury Hill covered by sheep.  The hill is brown compared to the surrounding countryside and I had to ask myself if the brownness is grass or some other plant that is seasonally brown, or if the hill is nearly denuded by the sheep (that tend to nibble down to the roots and kill the grass).  Without sufficient grass, this ancient monument will just wash away in the rain.  The photo in the book shows erosion gullies cascading down the sides of the mound.  (Maybe something has already been done to protect the mound from the sheep by now?  We can only hope so.  And perhaps the sheep are allowed there only briefly to provide some sort of ecological “mowing” function to keep tree saplings from taking over?)  Seems to me their sharp little hooves could dislodge bits of surface material.

 

An aerial view of a different pre-historic site in the DVD astonished me.  Some bureaucrat had put a paved road right across the Stonehenge site.  The circle of prehistoric stones there is just a small circle inside a much larger circle, which has a road across it.  Is nothing sacred?  Didn’t building the road disturb the archeological record?  I can’t imagine how a road could be built there without disturbing at least the top layers.  I was just aghast to see that road!  In their hubris they thought no doubt that they could find and knowledgeably analyze everything there was to find of any archeological significance before they sent in the bulldozers.  What a shame!!!  And the road had no purpose other than to get the tourists within closer walking distance of the stones. 

 

There is also a major highway that misses the Stonehenge outer circle by mere meters – what artifacts were lost putting in that highway?  Not much thought went into that either! 

 

There’s a major highway going right past Silbury Hill, too.  Not much thought went into that either!

 

I wonder what motivated the ancient people to keep on building Silbury Hill generation after generation for 400 years.  What was its purpose?  Did they climb it to make harvest offerings to fruitful Earth at the top of the hill?  Did they dance on it to celebrate the rising of the full moon?  Did they sing hymns to the nourishing Great Mother?  Sounds like more fun than sitting in front of a screen clicking on a mouse.

 

Sometimes a circle is just a circle.

Full-size image

 

It is interesting that the Silbury and Stonehenge design of a circle within a much larger circle is also found in the huge “rose windows” of many Christian cathedrals; for example, the beautiful rose window at the Cathedral of Notre Dame (for many more, search Google images for “rose window”).  Could it be we remember her still in a way, even after centuries of having spiritual traditions subverted to fit a male mold? 

 

And sometimes a circle is an octagon???  According to the UK’s English Heritage, their recent survey using satellite mapping “suggests . . . that the mound is not in fact truly circular: on the summit it appears to be more angular than circular, while at the base it is almost octagonal in form.”  That page also says that the steps may in fact be a spiraling ledge.  I went back and viewed the DVD again to see if the periphery looks octagonal.  Nope.  The aerial view in the DVD shows nearly the whole circumference.  Sure looks like it was intended to be a circle to me as the whole mound is rounded except for the top and crown.  I couldn’t find the particulars on their “suggestion of an octagon,” but I expect the edge is wobbly after 4,000 years of weathering and that’s all it is.

 

Of course there’s no way to prove (yet) what the Goddess Remembered DVD implies, that Silbury and Stonehenge were used for worship, let alone a particular kind of worship, but it’s a good bet, because worship of the Goddess was prevalent at that time.

 

Our legacy will not be an enigmatic pile of dirt, or the temples, art, and statues of the ancient Goddess worshippers.  Rather it will be gigantic piles of rusting containers oozing radioactive waste and hazardous chemicals, thousands of landfills overflowing with garbage, it will be a gazillion tons of crumbling asphalt and concrete, it will be a toxified and desertified landscape, polluted air and water, biological warfare germs, etc., the legacy of ripping-off the Earth.  We’d do better to get re-acquainted with our connection to the Earth.  A consumerist, hierarchical way of life is not the only way – there are alternatives that revere the Earth.  We hope.

 

-2008-

 

Slide show, music, and folders on my main page.

 

X Keywords:   Silbury Hill MSN Windows Live Spaces X

 

TRANSLATE THIS POST

 

TRANSLATE THIS POST

 

Interesting that MSN Windows Live Spaces has so many users.  One of their posts said, “115 million monthly Spaces users worldwide,” whatever that is.  And that was months ago – I don’t know how many now.

 

But think of that – so many people and so much potential for communication, potential for genuine, productive communication.  Think of the great possibilities for world peace.  Possibilities for any great thing that could evolve from this.  I am thrilled to have a global network and to be making Friends with people all over the world.  I have Friends now in many different countries.  I suspect that most of the MSN Spaces have only photos, no blog, but just the same, the potential is there.

 

Did you know you can translate the posts and comments that are written in something other than English?  Google provides a free translator for many languages.  To find it, go to http://www.Google.com, then click on “Language Tools.”  Paste the text to be translated in the “Translate Text” window, select the languages under that (the down arrows give you menus of language choices), and click on “Translate.”  Voilà!  By the way, this is how to paste:  first highlight the text by left clicking and dragging your cursor at the same time, then copy the highlighted text (press ctrl+c), then click where you want to paste, and then paste (press ctrl+v).

 

The translator will sort of translate.  Don’t expect to get much more than a general idea of what the topic is and maybe an outline of the text.  Language is so complex.  So much depends on how a word is used in a sentence.  So many words have a variety of meanings.  Translators can sometimes be inaccurate or fail to capture an essential idea.  So don’t expect too much.

 

The translator will help you read, but don’t expect that you will be able to start writing in a foreign language using a translator.  (In this post I am talking about online electronic translators, not the human variety.)

 

What if you don’t know what the language is?  In that case what I’ve done is copied a few of the foreign words and pasted them into Google and searched on them.  Then on the results pages, I look for clues like the name of a country, the name of a capital city or major city, a type of money or music or cooking, or words that might be similar in both English and the other language.  I look especially at the Web addresses as these are often in English.  When I click on results I look for national flags – many flags are identified near the end of the Google translator page.

 

Some languages look very similar to me, for example, Portuguese and Italian.  If you can narrow down the possibilities; for example, say it looks like a romance language, run the translator first for one, then another and see which gives mainly English as a result.

 

You can always leave a message at an MSN Space you are trying to translate and ask, “What language are you writing in?”  See if they can give you the answer.  The Profile for the Space may give a location or other clues. 

 

Of course if the original text is misspelled, the translator will choke on it.

 

And of course, if your text is private or confidential, don’t put it into the translator (unless you don’t mind sharing it with the owner of the translator).

 

What could be more intricate and complex than a language?

Full-size image

 

With Chinese, keep in mind that “simplified” characters are used in mainland China, and “traditional” characters are still used in some other places, like Taiwan.  It’s not easy to know which characters are which, but generally, the simplified look more simple.  That’s what I’ve been told.

 

If you want to test a translator, first translate from English to whatever language you are testing, let’s say Spanish.  Then copy and paste the resultant Spanish text into the translator and make it translate that into English.  See how closely it matches the English you started with.

 

I tried this with Google and a couple of other translator Web sites (all free), and here are the results:

 

Google translator: 

“I welcome you to my Web site and wish you a pleasant day.”

“Le doy la bienvenida a mi sitio Web y le deseamos un agradable día.”

“I welcome you to my Web site and wish you a pleasant day.”

 

PROMT translator:

“I welcome you to my Web site and wish you a pleasant day.”

“Doy la bienvenida a usted a mi Sitio Web y le deseo un día agradable.”

“I give the welcome to you to my Web site and wish him an agreeable day.”

 

SDL translator:

“I welcome you to my Web site and wish you a pleasant day.”

“Yo le doy la bienvenida a mi sitio web y le deseo un día agradable.”

“I welcome to my website and him desire a pleasant day.”

 

To find another translator online, type into Google “translate French” or whatever language you need translated.  Of course not all the results Google provides will be translators that actually work, and not all the results will be translators, so it takes a bit of patience.  When I did this, one supposed “translator” was some sort of threat that my virus protection software warned me not to open. 

 

Once you find a good translator, bookmark it in your Favorites, so you can quickly refer to it when visiting Friends in foreign places.  I’m not endorsing the translators on this page – you may be able to find a better one.

 

Some people write in English, even though it is not their first language, because they are practicing their English, or they want more visibility on the Web.  Since many languages are disappearing, I’d really prefer that people practice their own languages, and promote their own languages.  Sustain your own language!

 

Can you really practice your own language?  I’m still practicing at writing American English and it is my first language (and the only language I feel comfortable speaking or writing in).  I am still learning its complexities.  I think that written English is continually evolving, changing its form to more closely mimic spoken English which is also constantly evolving, but likely at a faster pace.  Think how much English has changed from the days of Shakespeare, hundreds of years ago – I’ll admit I can’t understand Shakespeare in the original.  When I hear teens talking, I just hope English doesn’t evolve so fast I can’t keep on learning it.

 

While the warmongers are bellowing at each other, thumping their chests, and circling each other over some oil field or gas pipeline or their “territory,” and ready to bomb, kill thousands, tens of thousands or more, and empty whole countries into refugee camps, the rest of us, who are not so impaired in our thinking, can devote our energies to improving communications, building communities (even online communities), enhancing cooperation, making the Earth greener and more child-friendly, engaging in civilized pursuits, learning what makes for a truly satisfying and fulfilling life, and making friends in foreign places.

 

I’ll guess MSN Windows Live Spaces supports at least 16 languages, looking at my “MSN help menu” when I search it for “language.”  Think how great that is, that a community is being supported and nourished here, despite language differences.  Each one can view the menus and module headings in MSN Spaces in their own language, even though the content written by the Space owner does not change.  For people from China visiting my Space, all my modules appear to have headings in Chinese (unless a visitor has selected English, etc. or unless I’ve changed the default heading).  Isn’t that wonderful!!!  By the way, the MSN ads are also language specific in case you wanted to know.

 

I know that because one day I inadvertently changed my Space so it appeared to me to be in Danish.  A link can contain a language command and if you click on a “Referring address” link on your Statistics page, you could suddenly be taking on the lingual preference of your visitor, unless MSN has made changes since I got in trouble that way.  Suddenly the ads, menus, and module headings in all of MSN Space were appearing to me in Danish!  Oh help!

 

I got curious as to how many people are actually blogging in English on MSN Windows Live Spaces.  I searched on the word “the” in their search engine (at the top of this page) and got about 1.64 million people, but only 403,000 blog entries  – which means how many people blogging in English?  Maybe “only” 5,000 or so (??) if each has about as many blog entries as I have.  Will I ever get to visit them all?  I doubt it.

 

Dear visitor, welcome!  I hope we can be Friends.

 

Dear Friends, thank you for being my Friend.

 

-2008-

 

Slide show, music, and folders on my main page.

 

X Keywords:   how to translate MSN Windows Live Spaces X

 

FREE TIBET

 

FREE TIBET

 

This week we remember the victims of the Chinese government:  the people of China denied democratic systems of government, and denied their human rights.

 

We remember especially the people of Tibet.  Until the communist takeover that began in 1949, Tibet had been politically independent for more than 1,400 years.  More than one million Tibetans died and 150,000 fled into exile, crossing the Himalaya to escape. 

 

Today, Tibetans in Tibet are second-class citizens in their own homeland.  Now the culture and language of Tibet are being deliberately destroyed by the Chinese government – monasteries infiltrated and strictly controlled, the people of Tibet denied access to their Teachers in exile, the countryside being overrun, via the new rail line, with transplanted Han Chinese gobbling up the land and resources of Tibet.  The capital, Lhasa, is now more than half Han Chinese.

 

In regard to its treatment of native people, the government of China is no more enlightened than the US government of past centuries which killed and persecuted Native Americans.  While history tells us that China was victimized by Western colonial powers, apparently that holds no lesson for the government of China, now emulating its former oppressors and now colonizing Tibet.  Is there some reason to treat others ethically and with compassion?  Do people have rights as specified in the United Nations Declaration on the Rights of Indigenous Peoples?

 

A concept of ever-living beauty

Full-size image

 

Why are the Chinese authorities afraid to let Buddhism flourish?  The Tibetans cannot even have a photo of their religious leader, the Dalai Lama, in their own homes.  To his great credit, the Dalai Lama continues to urge nonviolent conflict resolution.  This peaceful and peace-filled man is not a threat to anyone and yet the Chinese authorities are afraid of him, and refuse to cooperate.  When will we know the fate of the young Panchen Lama, kidnapped by the Chinese authorities when he was six years old? 

 

As China grows to be a major player in the world of business, maybe the Chinese government needs to re-evaluate the image it is presenting to its customers.  As a customer, I would say the image needs a lot of work.

 

I’ll add that this post is not China-bashing.  The Chinese people do not control their own government, despite the fact that the complete name of the country is “The People’s Republic of China.”  For the most part, those Chinese with Internet are denied full access, and presumably the vast majority of Chinese know only what their government wants them to know about Tibet.  This post focuses on the Chinese government, not China.  Governments everywhere need to be held accountable for their actions.

 

Also, I’m not advocating a return to a medieval system of government in Tibet.  I am advocating an end to persecution and a new, more autonomous future for Tibet.

 

I am informed in this post by Journey into Buddhism, Vajra Sky over Tibet, filmed secretly in Tibet, and 10 Questions for the Dalai Lama, both DVDs exceptional.  I’ll guess these DVDs would be illegal in China.  Now that I’ve posted on this touchy subject of Tibet and linked to the official Web site of the Dalai Lama, maybe my Space will be off-limits for servers in China, too.

 

The great jewel of Tibet will be lost unless the Chinese authorities can be persuaded to refrain from crushing it to bits.

 

Do not forget Tibet. 

 

Do not forsake Tibet. 

 

Let Tibet live.

 

Free Tibet.

 

-2008-

 

Slide show, music, and folders on my main page.

 

X Keywords:   Free Tibet MSN Windows Live Spaces X

 

IF YOU ARE REAL, PRESS ONE

 

IF YOU ARE REAL, PRESS ONE

 

Are there actually people out there who can’t wait until they get home to find out what e-mails they have?

 

I’m looking at these ads for gadgets.  Some little thingies you can hold in your hand or put in your purse to get driving directions, get your e-mail, and take photos when you’re out and about.

 

Even if I did have a longing for a camera in a thingy, I wouldn’t buy one just because I don’t like to read instruction manuals.  I don’t like to have to figure out how to program something.

 

Pity the parents who have to provide this stuff for their kids who just have to have it because “all the other kids have it.”

 

And when the gadgets break or don’t work right (and that is inevitable), then you can buy some more, and read some more instruction manuals.

 

I’d never want a “smart house,” where you have to program this and that.  I don’t want to have to read an instruction manual just to get my house to be a house.  I don’t want a thermostat with anything but a lever.  No little screen or buttons please.  Nor do I want a house wired with the latest (soon to be obsolete) technology.

 

Nor do I want a car with bells and whistles.  More bells and whistles just means more to repair.  When it comes time for a new car I’ll try to find one without a GPS and without a voice telling me, “You’ve made a wrong turn.”  I’ve never experienced that, but I’ve heard about these newfangled things.  I know how to use a map.  I don’t need a gadget giving me directions.  I wouldn’t trust a gadget to get it right.

 

If I were in charge of a phone company, I wouldn’t add any bells and whistles to the phones until they were actually capable of functioning as phones, that is, someone could actually hear the other party without a lot of static.  Why can’t I buy a phone that looks like a phone and actually works?

 

When I call the bank, I don’t want to hear “press 1 . . . press 2 . . . press 3 . . . press 4, etc.” through a half dozen menus, none of which say, “press zero if you are a real person who would like to talk to a real person.” 

 

Before humanity invents any more gadgets and “conveniences” that are either useless, annoying, or downright inconvenient, maybe we should make the trains run on time.  Maybe we should make sure everyone on the Planet has enough to eat. 

 

When something actually works, it can be wonderful.

Full-size image

 

I don’t want to program anything.  Not a phone.  Not a car.  Not a house.  Not a bank phone system.  Not a computer.  Not a DVD player.  Not anything.  I don’t like interacting with machines.  I don’t want to bond with these things.  Software is not something I want to spend time hassling with.  And maybe “program” is not the right word.  “Use menus”?  Whatever.  Even the copy machine at the copy machine store has a menu.  Even the gas pump has a menu.  Even the clock on my stove has a menu.  So often menus are not user-friendly. 

 

So far, I’ve managed to avoid using a cell phone even once (I don’t think people should be blasting their brain cells with microwave radiation).  How much longer can I hold out?

 

Now I admit that machines are essential to civilization, and if that’s your job to keep it all running smoothly, that’s a noble job.  But do machines have to be intruding into and ruling our everyday lives?  Calling the bank to check on some problem they created should not become a job for me.  After all, I am the customer.  They are supposed to be serving me, not making me work – “press 1, press 2, blah, blah, blah, and if you didn’t catch all that, press 9 to hear the menu again, and press pound to return to the main menu and start all over again.”  Shouldn’t the machine be invisible?  Or at least unobtrusive and undemanding?  Shouldn’t there be a limit on how many phone buttons I have to press?

 

Why can’t they design a bank phone answering system that has respect for the customer?  I just wait for something on the menu that maybe a machine could not deal with like, “If you have lost your card, press 8.”  Seriously.  That way maybe I can reach a real person.

 

The worst is the pseudo-human voice at the phone company ordering me to say “yes” or “no,” then I get disconnected five or ten times, because the real person who is being paid to answer the phone doesn’t feel like doing it, and when I call back again and again, I get the pseudo-human voice again and again.  I say “yes . . . . ,” “yes . . . . ,” “yes . . . . ,” into its voice recognition ears.   The one at the Internet service provider keeps saying something like, “I didn’t quite get that, could you repeat it?  Say ‘billing’ or say ‘technical support’.”  Just dreadful.  I try to keep my voice from sounding irritated.  After all, they are recording “for training purposes.” 

 

Has it come to this?  That I am now being ordered to speak by a machine?  How degrading!  Am I now the servant of the machine?  Bad enough to be ordered to press buttons.  Some people train their dogs to “speak” on cue.  Now a machine is training me.  For sure, the Queen of England has never experienced this indignity.  I suppose you are saying, “It’s alright to talk to an inanimate object, after all, on Star Trek, Captain Kirk talks to his starship’s computer all the time.”

 

I think that next time I am confronted by a pseudo-human telling me, “say ‘one’ or say ‘two’,” I will say, “fiddlesticks” or “go-fly-a-kite,” or simply “ah-ah-ah-ah.”  That will scramble it nicely and get me a real person on the line.

 

I have yet to find virus protection software that does not give me a conniption.  Of course the company doesn’t have technical support except something online, and if I’m having trouble installing their product, then of course I can’t go online while my computer is unprotected.  Guess how I spent the better part of Thursday.  (And it was only supposed to take five minutes.)

 

Now I find out that one of the local big box stores that sells electronics will do computer setup and diagnostics.  Well, it’s about time.  But will they do it right?

 

If I were in charge of everything, the trains would run on time.  There would be no hungry people.  There would be no global warming.  The phones would work.  Computers would work.

 

And no one but programmers would have to program anything.  No one would have to read instruction manuals, install anything, or press any buttons.

 

-2008-

 

Slide show, music, and folders on my main page.

 

X Keywords:   technology MSN Windows Live Spaces X

 

THE MARVELOUS SQUARE ROOT OF TWO

 

THE MARVELOUS SQUARE ROOT OF TWO

 

So many of the relationships in this design I’ve drawn involve the marvelous square root of 2. 

 

Full-size image

 

Most interesting I think is the fact that the ratio of the diameters of the circles, green circle to red, and red to white, is a constant.  It’s also interesting the way the values of the diameters add up, and the way they relate to the sides of the squares.  More on this later below.  I started a little journey here with my high school math.  Won’t you join me? 

 

Dimensions of a triangle given

In my graphic below, a diamond shape composed of four isosceles right triangles (45-90-45 degrees) forms a perfect square, set on end.  Let a = b = 1, where “a” is the altitude of a triangle and “b” is the base.  Using Pythagoras’ formula, we find that the third and longest side of the triangle, “c,” (the hypotenuse) is equal to √(a² + b²), that is, √2.

 

Full-size image with colors labeled (click here, then on the image)

 

Radius and diameter of a green circle

Next we find “r,” the radius of the green circle inscribed in the triangle, and also as we see later, the radius of the corner green circles.  See graphic 795611, where my formula for this is given as r = ab / (a + b + c) = 1 / (1 + 1 + √2) = 1 / (2 + √2) = (1 / (2 + √2)) x ((2 – √2) / (2 – √2)) = (2 – √2) / (4 – 2√2 + 2√2 – 2) = (2 – √2) / 2.  The diameter of a green circle is thus (2 – √2).  The diameter of the green circle can be expressed simply as a + b – c.

 

Radius and diameter of a red circle

The radius of a red circle equals a – r or b – r, that is, 1 – ((2 – √2) / 2) = (2 / 2) – ((2 – √2) / 2) = √2 / 2.  The diameter of a red circle is thus √2. 

 

Radius and diameter of the white circle

The radius of the enclosing or white circle equals a or b plus the radius of a red circle, that is, 1 + (√2 / 2) = (2 / 2) + (√2 / 2) = (2 + √2) / 2.  The diameter of the white circle is thus 2 + √2.

 

Circles are tangential

Two equal red circles, centered at diamond vertices, meet and are tangential at the midpoint of c, the hypotenuse.  It is clear that the diameter of a red circle, √2, equals c, also √2.  The red circles are tangential to the green circle (darkened) at the center of the design.  An inspection of the math confirms this: radius of red circle (√2 / 2) plus radius of green circle ((2 – √2) / 2) equals (√2 + (2 – √2)) / 2 = 2 / 2 = 1.  A similar inspection using the yellow lines parallel to a and b respectively, reveals that the corner green circle has a radius of (2 – √2) / 2 and is tangential to its neighboring red circles.  Thus the inscribed green circles and the corner green circles and center green circle are all equal.  The corner green circle is tangential to the white circle in that the sum of “d” (equal to and perpendicular to c) which is √2, plus the radius of a green circle ((2 – √2) / 2) equals (√2 x (2 / 2)) + ((2 – √2) / 2) = ((2 √2) / 2) + ((2 – √2) / 2) = ((2 √2) + (2 – √2)) / 2 = the radius of the white circle (2 + √2) / 2.

 

Ratios of diameters

The ratio of the diameters of the circles, green circle to red, and red to white, is a constant, (√2 – 1) to 1, a value which also happens to equal tan 22.5 degrees.  Here are the ratios calculated:  Green diameter to red diameter = (2 – √2) / √2 = ((2 – √2) / √2) x (√2 / √2) = (2√2 – 2) / 2 = √2 – 1.  Red diameter to white diameter = √2 / (2 + √2) = (√2 / (2 + √2)) x ((2 – √2) / (2 – √2)) = (2√2 – 2) / (4 – 2√2 + 2√2 – 2) = (2√2 –  2) / 2 = √2 – 1.  Of course the ratios of radii will follow the same pattern and have the same constant.

 

Radius of inscribed circle is confirmed

We can confirm the radius of the circle inscribed in the isosceles right triangle by constructing a line “h” from the left-most vertex (45 degree angle) through the center of the circle.  If the circle is inscribed and has the radius as calculated above, then this line will bisect the angle yielding an angle of 22.5 degrees.  Setting up an equation, tan of angle x = opposite / adjacent = green radius / red radius = ((2 – √2) / 2) / (√2 / 2) = (2 – √2) / √2 = ((2 – √2) / √2) x (√2 / √2) = (2√2 – 2) / 2 = √2 – 1.  A glance at a trig chart confirms this is indeed the tan of 22.5 degrees (0.4142). 

 

Sum of two diameters is 2

At some point I noticed that the sum of the diameter of a green circle plus the diameter of a red circle equals 2, when a = 1, as follows:  (2 – √2) + √2 = 2. 

 

Sum of two radii is 1

Naturally, the sum of the radius of a green circle plus the radius of a red circle equals 1, when a = 1, as follows:  ((2 – √2) / 2) + (√2 / 2) = ((2 – √2) + √2) / 2 = 2 / 2 = 1

 

Product of two diameters is 2

Later, someone pointed out to me that the product of the diameter of a green circle and the diameter of the white circle equals 2, when a = 1, as follows:  (2 – √2) x (2 + √2) = (4 + 2√2 – 2√2 – 2) = 2.  She got me started down this avenue more than a year ago by saying, “Did you realize a 45-90-45 degree triangle can have sides of 1 – 1 – √2?”

 

Diamond measures 2

I don’t know if it is worth mentioning that the height and the width of the diamond are also 2, when a = 1;  (1 + 1 = 2).  In other words, the diamond-square with sides of √2, has diagonals each equal 2.

 

Side of next square is 2

The sides of the next larger square equal 2.  This square is indicated by yellow lines in the graphic.

 

This design set to the next is ratio √2 to 1

Just as this design has circles associated with a square with sides of √2, there is a “similar” design (geometrically similar) associated with a square with sides of 1, formed by connecting in succession the midpoints of the previous square with sides of √2.  Just as the sides of these two squares are in a ratio of √2 to 1, the circle designs associated with these two squares are also in a ratio of √2 to 1.  Thus, the diameters of the circles in the first design (2 – √2, √2, and 2 + √2), when divided by √2, yield the following values for the diameters in the second design:  √2 – 1, 1, and √2 + 1.  These latter circles are yellow, blue, and lavender in my drawings above and below.

 

Full-size image

 

Diameter is the value of the constant

Notice that the diameter of the small circle (yellow) associated with the square of side 1, is √2 – 1, which equals the value of the constant (ratios of small to medium, medium to large diameters among the circles of a given square), and this value is also equal to the tan 22.5 degrees.

 

Diameter is the value of the reciprocal of the constant

The diameter of the large circle (lavender) in this design set, √2 + 1, is the reciprocal of the constant √2 – 1 (meaning 1 / (√2 – 1)), and thus represents the ratios of large to medium, medium to small diameters among the circles of a given square –  just the reverse order as paragraph above.  Therefore, the reciprocal √2 + 1 is also a constant.

 

Diameter is quotient of identical values

The diameter of the medium circle (blue) in this design set, 1, is the quotient of the diameter of the small circle (yellow) √2 – 1 and identical value, the constant √2 – 1.  Of course, any number divided by itself is 1. 

 

Diameter is quotient of other identical values

Another way of looking at this is that the diameter of the medium circle (blue) in this design set, 1, is the quotient of the diameter of the large (lavender) circle, √2 + 1 and the identical value, the reciprocal constant √2 + 1.  Of course, any number divided by itself is 1. 

 

Product of diameters is 1

The product of the diameter of the large (lavender) circle, √2 + 1, and the diameter of the small (yellow) circle, √2 – 1, equals 1, the diameter of the medium (blue) circle.  Of course, any number multiplied by its reciprocal is 1.  The calculation for this is (√2 + 1) x (√2 – 1) = (2 – √2 + √2 – 1) = 1.

 

Diameters and the sides of the squares

As shown in the graphic above, we have three squares with sides of 2, √2, and 1, each square nestled within the previous.  Notice that the values for the circle diameters in each design set are equal to either the difference of the sides of the squares, the sides of the squares, or the sum of the sides of the squares.  Each small circle in a design set is equal to the difference of the sides of two squares (2 – √2 and √2 – 1).  Each medium circle in a design set is equal to a side of a square (√2 and 1).  Each large circle in a design set is equal to the sum of the sides of two squares (2 + √2 and √2 + 1).

 

Area of design set is double that within

The area of a square is double the square within it.  Area of a square is a side squared, so for squares with sides of 2, √2, and 1, the areas are 4, 2, and 1; each double the square within.  Likewise, the areas of circles in a given design set are double the areas of circles in the next design set within it.  The area of a circle is Π x (radius)², so the areas of blue and red circles are 1/4 Π and 1/2 Π respectively,  latter twice the former.  While dimensions differ by a factor of √2 between design sets, the areas differ by a factor of 2.

 

Design dimensions double

With every second iteration of the design (factor √2 x factor √2) the design dimensions double.

 

A circle with a diameter of 2

If I drew the next larger design set, the medium circle in that set (red-similar) would have a diameter of 2.  This diameter would be the diameter of the red circle (√2) multiplied by the enlargement factor √2, or (√2)².  A diameter of 2 can also be thought of as the product of the red diameter times the red diameter.  A diameter of 2 can also be thought of as the sum of two blue circle diameters (1 + 1 = 2).

 

Intersection of three circles

The center of a small (yellow) inscribed circle is at the intersection of a red circle and the center green circle.  The calculation for this:  by inspection, the distance from the center of the red circle to the boundary of the square with side of 1 is 1/2.  Add to this the radius of the yellow circle, √2 – 1.  The sum should be √2/2, the radius of the red circle, and it is:  (1/2) + ((√2 – 1) / 2) = (1 + (√2 – 1)) / 2 = √2 / 2.

 

Concentric circles

The corner yellow circles are concentric with the red circles.

 

Diameters equal to trigonometric values

Other nice relationships in this design:  both sin and cos of 45 degrees equal 1 / √2, or put another way, 1 / √2  = (1 / √2) x (√2 / √2) = √2 / 2, the radius of a red circle.  The tan of 45 degrees is 1, the diameter of a blue circle.  As mentioned earlier, the diameter of a yellow circle, √2 – 1, is the tan of 22.5 degrees.

 

See how the diameters add up

I decided to get a bit more methodical in looking at sums and so made the following table which shows the diameter values of two design sets, designated greater and lesser.

 

Diameter values of two design sets
Relative diameter size	Left	Small	Medium (center value)	Large	Right
Greater set	3√2 – 4	2 – √2 green	√2 red	2 + √2 white	3√2 + 4
Lesser set	3 – 2√2	√2 – 1 yellow	1 blue	√2 + 1 lavender	3 + 2√2

 

Rule 1:  The sum of any two consecutive diameters in a column equals the value in the next column for the lesser design set.  For example:

>> green (2 – √2) + yellow (√2 – 1) = blue (1), and

>> red (√2) + blue (1) = lavender (√2 + 1)

 

Rule 2:  The sum of any two consecutive diameters in a row equals the value in the column above the right-side figure.  For example:

>>  yellow (√2 – 1) + blue (1) = red (√2), and

>>  blue (1) + lavender (√2 + 1) = white (2 + √2)

 

Rule 3:  Notice that values on either side of the “center value” make up matched pairs that differ only in sign.  In the lesser design set, the product of each matched pair is 1.  In the greater design set, the product of each matched pair is 2.  For example:

>> (√2 + 1 ) x (√2 – 1) = (2 – √2 + √2 – 1) = 1.

>> (3√2 – 4) x (3√2 + 4) = (18 + 12√2 – 12√2 – 16) = 2

 

I got out pencil and paper and my guess is that this progression continues with products developing in succession, each a factor of two greater than the previous:  1, 2, 4, 8, 16 . . . etc.

 

One more sum and product

Both the sum and the product of red and white diameters equal the value of the large circle (2√2 + 2) in the next greater design set.

 

I suppose there could be even more interesting relationships in these designs, but that’s enough for now.

 

-July 19, 2010-

-updated August 1, 2010-

 

Slide show, music, and folders on my main page.

 

My related drawings:

 

794206 Each circle 2X diameter of two before, factor square root 2 over previous, 2X area of previous

 

795805 Each square 2X width of two before, factor square root 2 over previous, 2X area of previous

 

795811 Wreath circle diameter is the difference of the sides of the squares (square root  2 – 1) (circles in triangles oriented differently from above to form wreaths in this and next two drawings)

 

795811b Graphic for diameter is the difference of the sides of the squares (square root 2 – 1)

 

795812 Outer wreath circle diameter is the difference of the sides of the squares (2 – square root 2), ratio diameter to diameter is square root 2

 

796207 Five equal circles

 

796219 Design set added twice dimensions of first set

 

MANIFESTATION OF THE GOLDEN MEAN

 

MANIFESTATION OF THE GOLDEN MEAN

 

The golden mean can be found among circles associated with a pentagram and pentagon.

 

In the following drawing, there is a pentagram (five-pointed star), overlaid on a pentagon (five-sided polygon), within an enclosing circle.  The lines of the pentagram and pentagon form identical triangles within which are inscribed 6 equal circles:  a central circle and a wreath of five circles.  The central circle fills the inverted small pentagon within the pentagram and is tangential to all five sides there.  Each of the wreath of five circles is tangential to two sides of the pentagon and a “longest” pentagram line.  An examination of the angles in the design shows that a side of the pentagon is equal to a “short” plus a “medium” segment of the pentagram, thus the central circle inscribed in a triangle composed of two sets of these segments and a “longest” pentagram line is equal to the other five circles.

 

Pentagram, pentagon in circle with six equal circles, radius to diameter is 1.618 . . . golden mean.

Full-size image

 

While drawing this design, I questioned whether the radius of the design’s enclosing circle could be related to the diameter of the six equal circles by a ratio of 1.6180339887498948482045868343646 to one, in other words, the “golden mean.”  I found that is the case.

 

Many people know that the golden mean can be found in the relationship among the various sized segments of the pentagram, as follows:  medium segment to short segment, long to medium (where “long” equals a short plus a medium segment), longest to long, all these ratios equal 1.6180339887498948482045868343646 to one, in other words, the “golden mean.”  Many people also know the golden mean can also be found in the Fibonacci Series.  Each number in this series is added to the previous number to yield the next; for example, 1, 1, 2, 3, 5, 8, 13, 21 . . . . 6765, 10946, 17711, etc.  As the series progresses, the quotients of successive numbers become closer and closer to the golden mean.  For example, the quotient of the latter two numbers is 1.6180339850173579389731408733784, matching to eight decimal places.  (How is it that this golden mean number is found in two disparate places?)

 

Here are calculations, using simple high school math, to show that the radius of the enclosing circle and the diameter of a wreath circle are in the ratio of 1.6180339887498948482045868343646 to one, in other words, the “golden mean.” 

 

Full-size image

 

Using the diagram above,

 

AB = 1.6180339887498948482045868343646, when AC = 1.0 (longest segment of pentagram design to its long segment).

 

AD = 1/2 AB =  1.6180339887498948482045868343646 / 2 = 0.809016994374947424102293417182

 

AE = AD (because line AF, to the center of circle F, bisects angle CAD, forming two equal sub-triangles, triangle DAF and triangle EAF).

 

Right triangle CEF is formed by drawing radius EF, perpendicular to side AC (because EF is parallel to DG, a line drawn from the right angle to be perpendicular to AC, the hypotenuse of right triangle ADC, similar triangles have been formed and triangle CEF is similar to triangle ADC).

 

Find CD, the remaining side of triangle ADC:

 

An examination of angles in the design shows that

sin angle CAD = sin 36°= opposite / hypotenuse = CD / AC = CD / 1.0 = 0.58778525229247312916870595463907

CD = 0.58778525229247312916870595463907

 

Find radius EF, using proportions in similar triangle ADC.

 

EF / CE = AD / CD

First find CE:

CE = AC – AE

CE = 1.0 – 0.809016994374947424102293417182 = 0.190983005625052575897706582818 

Then, AD / CD = 0.809016994374947424102293417182 / 0.58778525229247312916870595463907= 1.3763819204711735382072095819097

Therefore, EF /CE = 1.3763819204711735382072095819097

EF = 1.3763819204711735382072095819097 x CE

EF = 1.3763819204711735382072095819097 x 0.190983005625052575897706582818  = 0.26286555605956680301283454242483

Diameter is twice radius

2 x 0.26286555605956680301283454242483 = 0.52573111211913360602566908484966

 

Now that we have the diameter of a wreath circle (0.52573111211913360602566908484966), next find the radius of the enclosing circle.

 

In triangle CKL, L is the center of the enclosing circle, K is the midpoint on AC (CK = 1/2 AC = 0.5), and CL is the hypotenuse of triangle CKL and also the radius of circle L.  An examination of the design shows that angle CLK is 36 degrees.

 

Sin 36°= opposite / hypotenuse = CK / CL = 0.5 / CL = 0.58778525229247312916870595463907

 

CL = 0.5 / 0.58778525229247312916870595463907 = 0.85065080835203993218154049706312

 

Divide radius CL (0.85065080835203993218154049706312) by diameter of circle F (0.52573111211913360602566908484966). 

 

The result is 1.6180339887498948482045868343622, the golden mean!!!  (Some erosion of the last few decimal places can be expected from the calculator for all the numbers in this post.)

 

Well, I thought that was it.   Then I noticed that the distance from a pentagram crisscross to the circumference of the enclosing circle (distance MN in the diagram), seemed to equal the diameter of a wreath circle.  And it does!  Here are the calculations:

 

An examination of the design shows that the distance MN equals CM, the diameter of the enclosing circle, minus twice the equivalent of CD.  Notice that CD = DN, and the diameter of the enclosing circle is twice the radius CL.

 

MN = 2CL – 2CD

MN = (2 x  0.85065080835203993218154049706312) – (2 x 0.58778525229247312916870595463907)

MN = 1.701301616704079864363080994126 – 1.175570504584946258337411909278

MN = 0.525731112119133606025669084848, same as the diameter of a wreath circle obtained above.

 

Thus the design can be redrawn below, again with 6 equal circles, but with the wreath of five anchored on the circumference of the enclosing circle, each wreath diameter spanning the distance between a pentagram crisscross and the circumference of the enclosing circle, highlighting the relationship there, that is, the golden mean:  LM / MN = 1.6180339887498948482045868343652

 

Pentagram, pentagon in circle with six equal circles, radius to diameter is 1.618 . . . golden mean.  Wreath circles intersect pentagram crisscross and circumference of enclosing circle.

Full-size image

 

So isn’t it nice to find the golden mean among circles associated with the pentagram and pentagon?

 

Posted on April 20, 2010

 

Slide show, music, and folders on my main page.